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UNIVERSITY  OF  CALIFORNIA 

ANDREW 

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A  TREATISE 


ON 


HYDRAULICS 


BY 


HENRY    T.    BOVEY, 

M.  INST.  C.E.,  LL.D.,  F.R.S.C., 

Professor  of  Civil  Engineering  and  Applied  Mechanics, 
McGill  University,  Montreal. 


SECOND    EDITION,    REWRITTEN. 
FIRST    THOUSAND. 


NEW   YORK: 

JOHN   WILEY   &   SONS. 
LONDON:    CHAPMAN  &    HALL,    LIMITED. 

I9O  I  o 


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, 


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Copyright,  1895,  1901, 

BY 
HENRY  T.    BOVEY. 


HALLIDIE 


flOBERT    ORUMMONO,    PRINTER      NEW   YORK. 


PREFACE. 


THE  present  treatise  is  the  outcome  of  lectures  delivered 
in  McGill  University  during-  the  last  ten  or  twelve  years,  and 
although  intended  primarily  for  the  use  and  convenience  of  the 
student  of  hydraulics,  it  is  hoped  that  it  may  also  prove 
acceptable  to  the  engineer  in  general  practice. 

In  order  to  render  the  treatment  qf  the  subject  more  com- 
plete, free  reference  has  been  made  to  standard  authors  on  the 
subject.  The  examples  introduced  to  illustrate  the  text  have 
also  been  selected  in  part  from  the  works  of  such  well-known 
writers  as  Weisbach,  Osborne  Reynolds,  and  Cotterill,  but 
the  greater  number  are  such  as  have  occurred  in  the  course  of 
the  author's  own  experience.  The  tables  of  coefficients  of 
discharge  have  been  prepared  from  the  results  of  experiments 
carfied  out  in  the  Hydraulic  Laboratory  of  the  University. 
These  experiments  are  still  being  continued  and  may  probably 
form  the  subject  of  a  special  paper. 

The  author  desires  to  acknowledge  many  suggestions 
offered  by  Mr.  Bamford,  and  to  express  his  deep  obligation 
to  Professor  Chandler  for  much  labor  and  time  given  to  the 
revision  of  proof  sheets. 

HENRY  T.  BOVEY. 

MONTREAL,  November,  1895. 

iii 


PREFACE  TO  SECOND   EDITION. 


THE  present  edition  of  the  work  on  "Hydraulics"  has 
been  practically  rewritten,  the  various  chapters  having  been 
rearranged  and  in  some  cases  completely  altered  in  order  to? 
allow  of  necessary  corrections  and  of  the  introduction  of  muclu 
new  matter. 

In  Chapter  I,  articles  on  the  whirling  and  rotation  of  fluids 
have  been  inserted,  the  article  on  "  Weirs  and  Notches  "  has 
been  completely  rewritten,  and  there  has  been  added  a  resume 
of  Bazin's  experimental  work  on  weirs,  a  complete  account  of 
which  appears  in  the  Annales  des  Fonts  et  Chaiissees. 

In  Chapter  II  will  be  found  a  large  amount  of  new  material,, 
including  the  results  of  experiments  collaborated  and  tabulated 
by  Mr.  C.  W.  Tutton  of  Buffalo,  to  whom  I  also  owe  many 
thanks  for  various  useful  suggestions  and  for  the  graphical 
representation  of  the  results  of  the  pipe-flow  experiments. 

Chapter  III  has  been  considerably  changed  and  lengthened. 
The  results  of  the  experiments  by  Bazin,  Ganguillet  and 
Kutter,  and  others  are  given  in  detail  and  tables  giving  the 
values  of  the  constants  in  the  several  standard  formulae,  both 
in  English  and  metrical  units,  are  added  at  the  end  of  the 
chapter. 

Chapter  IV  contains  new  articles  on  "accumulators, 
presses,  and  water-engines. ' ' 

Chapter  V  has  been  completely  rewritten  and  now  includes 


96051 


vi  PREFACE    TO  SECOND  EDITION. 

a  discussion  of  the  analysis  of  the  impact,  Borda,  centrifugal, 
and  other  turbines. 

Chapters  VI,  VII,  and  VIII  in  the  new  volume  replace 
Chapter  VII  of  the  old  volume.  Chapter  VI  deals  exclusively 
with  water-wheels;  Chapter  VII  contains  new  matter  and 
treats  of  the  various  classes  of  turbines  which  have  not  been 
dealt  with  in  Chapter  V.  Chapter  VIII  is  entirely  new  and 
deals  with  centrifugal  pumps.  Much  of  the  information  incor- 
porated in  this  chapter  has  been  obtained  through  the  kindness 
of  Mr.  A.  F.  Hall  of  Boston,  who  has  given  valuable  hints  and 
suggestions  and  who  has  also  furnished  important  practical 
examples. 

It  is  hoped  that  the  large  amount  of  new  material  and  the 
various  tables  which  have  been  added  to  this  volume  will 
indicate  the  progress  which  is  being  made  in  reducing  the 
subject  of  Hydraulics  to  an  exact  science  and  that  these  addi- 
tions, more  especially  the  tables,  will  add  considerably  to  the 
usefulness  of  the  book  for  the  purposes  of  the  practical  engineer. 

I  have  now  only  to  express  my  gratitude  to  my  colleague, 
Dr.  Coker,  for  suggestions  made  from  time  to  time  and  for  his 
great  kindness  in  revising  the  proof  sheets. 

HENRY  T.  BOVEY. 

October,  1901. 


CONTENTS, 


CHAPTER   I. 

GENERAL  PRINCIPLES,    FLOW    THROUGH    ORIFICES,    OVER    WEIRS,  ETC. 

PAG* 

Fluid-motion x 

Steady  motion T 

Permanent  regime I 

Stream-line  motion 2 

Motion  in  plane  layers 2 

Laminar  motion 2 

Density 3 

Compressibility 5 

Head   , 7 

Continuity 7 

Bernoulli's  theorem 8,  108 

Applications  of  Bernouilli's  theorem 12 

Rotation  of  a  fluid I7 

Whirling  fluids xg 

Orifice  in  a  thin  plate « 22 

Torricelli's  theorem 24 

Flow  through  orifices  in  vessels  in  motion 26 

Flow  in  frictionless  pipe 27 

Hydraulic  coefficients 29 

Tables  of  coefficients  of  discharge   39 

Miner's  inch , 44 

Inversion  of  jet 48 

Time  of  emptying  or  filling  a  lock , 50 

General  equations 53 

Loss  of  energy  in  shock 55 

Mouthpieces 58 

Energy  and  momentum  of  a  jet 69 

Radiating  current 70 

vii 


Vlil  CONTENTS. 

PAGB 

Vortex  motion 74 

Large  orifices  in  vertical  plane  surfaces 78 

Notches  and  weirs 83 

Reservoir  sluices 97 

Bazin's  flow  over  weirs 99 

Examples , 109 


CHAPTER    II. 

FLUID-FRICTION    AND    PIPE-FLOW. 

Fluid  friction Ut 

Laws  of  fluid-friction 123 

Surface-friction  of  pipes 126 

Darcy's  results 127 

Reynolds'  results 129 

Critical  velocity 129 

Poiseuille's  results 130 

Resistance  of  ships 131 

Pipe-flow  assumptions 132 

Steady  flow  in  pipe  of  uniform  section 133 

Influence  of  pipe's  inclination  on  flow 138 

Formulae  of  Darcy,  Hagen,  Thrupp,  Reynolds,  etc 139 

Diagrams  of  pipe-flow 146 

Values  of  c ,  x,  a.ndy  in  the  formula  v  —  cmxiy 153 

Transmission  of  energy 156 

Pressure  due  to  shock 160 

Flow  in  uniform  pipe  connecting  two  reservoirs 162 

Losses  of  head  due  to  abrupt  changes  of  section,  elbows,  valves,  etc..    164 

Nozzles 174 

Ellis'  experiment  on  nozzles 177 

Freeman's  nozzle  experiments 178 

Motor  driven  by  water  from  a  pipe 179 

Siphons 181 

Inverted  siphons 182 

Air  in  a  pipe 183 

Flow  in  a  pipe  of  varying  diameter 184 

Equivalent  uniform  main 186 

Branch  mains  of  uniform  diameter 188 

Flow  in  pipes  leading  from  three  reservoirs  to  a  common  junction. . . .   191 

Mains  with  any  required  number  of  branches 201 

Variation  of  velocity  in  a  transverse  section 202 

Gauging  of  pipe-flow 207 

Examples 210 


CONTENTS.  IX 

CHAPTER   III. 

FLOW   OF   WATER   IN    OPEN    CHANNELS. 

PACK 

Channel-flow  assumptions 220 

Steady  flow  in  channels  of  constant  section 221 

Retarding  effect  of  air,  etc 224 

Table  of  slopes  and  mean  velocities  of  flow 227 

Form  of  channel  cross-section 228 

Best  dimensions  for  trapezoidal  channel 233 

Aqueducts 240 

Formulae  of  Prony,  Eytelwein,  Beardmore,  and  Tadini 247 

Bazin's  formulae 249 

Table  of  values  of  a  and  ft 249 

Table  of  values  of  y 250 

Ganguillet  &  Kutter's  formula 250 

Table  of  values  of  n 251 

Formulae  of  Manning, .Tutton,  Humphreys  &  Abbott,  Gankler 252 

Variation  of  velocity  in  channel  cross-section 257 

Boileau's  formulae 268 

Tables  of  erosion  and  viscosity 266 

River-bends 269 

Channels  of  varying  cross-section 271 

Tables  of  values  of  /for  standing  wave 281 

Longitudinal  profile  and  Ruhlmann's  law 285 

Channel  of  rectangular  section  of  small  slope 287 

Channel  of  great  width  as  compared  with  the  depth 288 

Tables  of  values  of  backwater  function,  <p  (z) 290,  292 

Change  of  section 293 

Gauging  of  streams  and  water-courses 297 

Determinations  of  mean  velocity  of  flow 298 

Tables   of    values    of  coefficients   in  formulae  of  Bazin,  Ganguillet  & 

Kutter,  and  Manning 311-327 

Examples 328 

CHAPTER   IV. 

RAMS,   PRESSES,    ACCUMULATORS,  WATER-PRESSURE    ENGINES. 

Hydraulic  rams 334 

Packing 336 

Hydraulic  press 337 

Hydraulic  jack 337 

Punching  bear , .  339 

Accumulators 340 


x  CONTENTS. 

PAGff. 

Differential  accumulators 342 

Steam-accumulator 344 

Hydraulic  engines 344 

Losses  of  energy  in  hydraulic  engines 351 

Brakes 353 

Examples 355, 


CHAPTER   V. 

IMPACT,   REACTION,   IMPACT  AND  TANGENTIAL  TURBINES. 

Impact  upon  a  flat  vane 359 

"          "      a  series  of  flat  vanes 362; 

"          "     surface  of  revolution 364 

"  "      a  series  of  surfaces 366 

"a  bordered  vane 368 

Impact  apparatus 369 

Coefficient  of  impact 372 

Reaction 373 

Jet  propeller 373 

Jet  reaction  wheel  (Scotch  turbine) 375 

Impact  wheel 378 

Borda  turbine 382 

Effect  of  friction  on  impact  turbine 384 

Danaldes 386 

Tub-wheel 387 

Impact  on  a  curved  vane 388 

Tangential  (centrifugal)  turbines 393 

Jet  turbine 400 

Resistance  to  motion  of  a  solid  in  a  fluid 402 

Pressure  on  a  thin  plate 404 

Pressure  on  a  cylindrical  body 406 

Examples '. 408 


CHAPTER    VI. 

VERTICAL    WATER-WHEELS. 

Classification 416 

Undershot  wheels 416 

Wheels  in  straight-race 418 

Losses  in  straight-race 421 

Mechanical  effect  of  straight-race -420,  423 

Poncelet  wheel 424. 


CONTENTS.  xi 

PACK 

Mechanical  effect  of  Poncelet  wheel 428 

Efficiency  of  Poncelet  wheel 428,  432 

Form  of  buckets 435,  458 

Sluices 437 

Breast- wheels 440 

Speed  of  wheels 441 

Mechanical  effect  of  wheels 442 

Sagebien  wheel 449 

Overshot  wheels 450 

Velocity  of  wheels „ (. . .   450 

Effect  of  centrifugal  force  in  overshot  wheel 451 

Weight  of  water  on  wheel 452 

Arc  of  discharge 452,  457 

Capacity  of  bucket. . . . 458 

Useful  effect  of  overshot  wheel 467 


CHAPTER  VII. 

TURBINES. 

Reaction  and  impulse  turbines 482 

Actual  path  of  a  fluid  particle  in  a  turbine 486 

Classification  of  turbines 490 

Analysis  of  turbine 497 

Practical  coefficients 519 

Theory  of  draft  tubes 529 

Losses  and  mechanical  effect 531 

Examples 539 

I  CHAPTER   VIII. 

CENTRIFUGAL    PUMPS. 

General  statement 547 

Analysis  of  centrifugal  pump 553 

Losses  in  hydraulic  resistance    554 

Blade-angles 556 

Volute 558 

Whirlpool  chamber 565 

Practical  coefficients 569 

Examples 572 


HYDROSTATIC   PRINCIPLES. 


FUNDAMENTAL  PRINCIPLES  OF  HYDROSTATICS.— .F/w/i/s-  may  be 
divided  into  two  classes  : 

Liquids,  which  are  incompressible,  or  nearly  so.  showing  no  sensible 
change  of  volume  under  changes  of  pressure,  and 

Gases,  which  are  compressible,  changing  in  volume  with  changes  of 
pressure. 

The  pressure  of  a  perfect  fluid  on  any  surface  with  which  it  is  in  con- 
tact is  perpendicular  to  the  surface. 

The  pressure  of  a  fluid  at  any  point  of  a  surface  is  the  pressure  per 
unit  of  area. 

The  pressure  at  any  point  of  a  fluid  is  the  same  in  every  direction. 

Any  pressure  applied  to  the  surface  of  a  fluid  is  transmitted  equally 
to  all  parts  of  the  fluid. 

The  density  of  any  uniform  substance  is  the  mass  of  a  unit  of  volume 
of  the  substance. 

The  intrinsic  weight  of  a  substance  is  the  weight  of  a  unit  of  volume 
of  the  substance,  expressed  in  terms  of  some  standard  unit  of  weight. 
The  difference  in  the  unit  due  to  change  of  locality  is  very  slight,  the 
ratio  of  polar  to  equatorial  gravity  being  32.2527  :  32.088. 

The  specific  gravity  of  a  substance  is  the  ratio  of  the  weight  of  a,ny 
volume  of  the  substance  to  the  weight  of  an  equal  volume  of  a  standard 
substance. 

If  fluid  volumes  V,  V,  V" — of  densities  p,  p',  p" — are  mixed  together, 
the  density  of  the  mixture  =  2(p  V)  •+-  2(  V). 

If  fluid  volumes  V,  V,  V"—o(  specific  gravities  s,  s',s" — are  mixed 
together,  the  specific  gravity  of  the  mixture  =  ^(sV)  -*-  ^(V). 

The  pressure  in  a  homogeneous  fluid  at  rest  under  gravity  increases 
uniformly  with  the  depth,  or.  in  other  words,  the  difference  of  the  pres- 
sures at  any  two  points  varies  as  the  vertical  distances  between  the 
points. 

xiii 


xiv  HYDROSTATIC  PRINCIPLES. 

Analytically,  the  difference  of  pressure  =  u>z,  w  being  the  intrinsic 
weight  of  the  fluid  and  2  the  difference  of  level. 

The  free  surface  of  a  liquid  at  rest  under  gravity  is  a  horizontal  plane.. 

The  common  surface  of  two  liquids  of  different  densities,  which  do 
not  mix,  is  a  horizontal  plane,  when  at  rest  under  gravity.  If  a  number 
of  liquids  of  different  densities,  e.g.,  mercury,  water.and  oil,  are  poured 
into  a  vessel,  they  will  come  to  rest  with  their  common  surfaces  horizon- 
tal planes,  the  densities  of  the  liquid  increasing  downwards. 

The  surfaces  of  equal  pressure  are  horizontal  planes. 

The  pressure  of  a  liquid  on  any  horizontal  area,  A,  is  equal  to  the 
weight  of  a  column  of  the  liquid  of  which  the  area  is  the  base  and  of 
which  the  height,  z,  is  equal  to  the  depth  of  the  area  below  the  surface, 
i.e.,  wAz  (disregarding  the  pressure  on  the  free  surface). 

The  whole  pressure  of  a  fluid  on  a  submerged  surface  is  the  sum  of 
all  the  normal  pressures  exerted  by  the  fluid  on  every  portion  of  the  sur- 
face and  (disregarding  the  pressure  on  the  free  surface)  is  equal  to  the 
weight  of  a  column  of  liquid  of  which  the  base  is  equal  to  the  area  of 
the  surface,  and  the  height  is  equal  to  the  depth  of  the  centroid  of  the 
surface  below  the  surface  of  the  liquid.  Thus  : 

(a)  The  total  normal  pressure  on  a  wall  of  width  b,  sloping  at  6  to 
the  vertical  and  retaining  water  which  rises  over  a  length  z  of  the  wall 

z  ivb2*  cos  0 

=  ivbz  —  cos  0  —  ---  ' 

(b)  The  total  pressure  on  a  circular  valve  of  diameter  d,  with  its  cen- 

TtlP 

troid  z  below  the  surface  =  w  —  z. 

4 

(0  The  total  normal  pressure  on  a  lock-gate  of  width  b  and  on  which 
the  water  rises  to  a  height  z  =  wbz^  —  —ivbz'1. 

The  pressure  between  a  pair  of  lock-gates  =  pressure  on  the  hinge 
post  =  ±wbz*  sec  a,  ia  being  the  angle  between  the  gates. 

'The  centre  of  pressure  of  a  plane  area  is  the  point  of  action  of  the 
resultant  fluid-pressure,  (/?)•  upon  the  plane  area. 

If  y,  ~z  are  the  horizontal  and  vertical  distances  of  the  C.  of  P.  from 
the  vertical  and  horizontal  axes  through  the  C.  of  G.  of  the  area, 

_  wD        iv  D  _     D  -  _  ool  _  9 

~~  ~X~  =  ^Ah  =  ~Ah  '  a  ~~  R    ~ 


D  being  the  product  of  inertia  about  the  axes  ;  /the  moment  of  inertia 
of  the  area  about  the  axis  of  y  ;  //  the  depth  below  the  surface  of  the 
centroid,  and  k  the  radius  of  gyration. 


HYDROSTATIC  PRINCIPLES.  xv 

Ex.  i.  Depth  of  C.  of  P.  of  a  parallelogram  with  one  edge  in  surface 
.=  f  of  depth  of  opposite  edge. 

Ex.  2.  Depth  of  C.  of  P.  of  a  triangular  area,  the  middle  points  of  the 

sides  being  at  depths  d\,  di,  d*  below  the  surface,  =  —  -  -  --  —  , 

tfi    +  rta    +    a* 

and  (a)  if  vertex  is  in  surface  and  base  horizontal,  depth  =  f  of  depth  of 

base  ; 
•     (#)  if  base  is  in  surface,  depth  =  |  of  depth  of  vertex  ; 

(r)  i/  vertex  is  in  surface  and  y  and  z  are  depths  of  ends  of  base,  the 

i  y  —  23 
depth  W^-y. 

The  resultant  pressure  on  the  surface  of  a  solid,  wholly  or  partially 
immersed  in  a  fluid,  is  equal  to  the  weight  of  the  displaced  liquid  and 
acts  vertically  upwards  in  a  line  passing  through  the  centroid  of  the  dis- 
placed liquid.  In  other  words,  a  solid  immersed  in  a  liquid  appears  to 
lose  as  much  of  its  weight  as  is  equal  to  the  weight  of  the  fluid  it  displaces. 

If  a  homogeneous  body  float  in  a  liquid,  its  volume  will  bear  to  the 
volume  immersed  the  inverse  ratio  of  the  specific  gravities  of  the  solid 
and  liquid. 

A  body  of  weight  W,  carrying  a  load  P,  floats  in  a  liquid,  G  and  h 
being  the  centres  of  gravity  of  the  body  and  of  the  displaced  water,  so 
that  GH  is  vertical.  If  the  load  P  is  shifted,  the  body  will  heel  through 
an  angle  0  and  the  point  H,  also  called  the  centre  of  buoyancy,  will 
move  on  a  curve  or  surface  of  buoyancy  to  a  new  position  H'  ,  the  line 
G'H  connecting  H'  with  the  new  position  of  the  C.  of  G.  of  the  body 
being  vertical.  If  6  is  small,  the  ultimate  position  of  M,  the  intersec- 
tion of  HG  and  H'  G'  ,  is  called  the  metacentre,  and  M  is  therefore  the 
centre  of  curvature  of  the  surface  of  buoyancy  at  H.  For  stability  of 
equilibrium  M  must  be  above  G.  Theoretically, 


iv  I      wAk*       A& 

•       HM  —  -TJJ-  =   -  =7-    =    —  -  -  , 

W        wV  V 

A  being  the  water-line  area  and  F  the  volume  of  liquid  displaced  by 
body. 

CAPILLARY  PHENOMENA.—  If  a  glass  tube  of  fine  bore  is  placed  verti- 
cally in  a  liquid  like  water,  which  wets  the  glass,  the  water-surface  on 
the  outside  next  the  glass  is  elevated  and  slightly  concave,  while  on  the 
inside  the  water-surface  is  concave  and  there  is  a  marked  elevation  above 
the  outside  surface. 

With  a  liquid  which  does  not  wet  the  glass,  like  mercury,  an  opposite 
effect  is  observed.  There  is  a  depression  on  the  outside  and  the  surface 


xvi  HYDROSTATIC  PRINCIPLES. 

is  slightly  convex,  while  on  the  inside  the  surface  is  convex  and  there  is 
a  marked  depression  below  the  outside  surface. 

SURFACE  TENSION. — At  the  bounding  surface  separating  air  from 
any  liquid,  or  between  two  liquids,  there  is  a  surface-tension  which  is, 
the  same  at  every  point  and  in  every  direction. 

At  the  line  of  junction  of  the  bounding  surface  of  a  gas  and  a  liquid 
with  a  solid  body,  or  of  the  bounding  surface  of  two  liquids  with  a  solid 
body,  the  surface  is  inclined  to  the  surface  of  the  solid  body  at  a  definite 
angle,  depending  upon  the  nature  of  the  solid  and  the  liquids. 

The  surface-tension  is  independent  of  the  curvature  of  the  surface 
but,  if  the  temperature  be  increased,  it  diminishes. 


USEFUL  CONSTANTS. 


The  following  abbreviations  are  used:  Metre  =  m.;  sq.  metre  =  m.*; 
cubic  metre  =  m.3;  centimetre  =  cm.;  sq.  centimetre  =  cm.1;  cubic  centi- 
metre —  cm.3;  kilometre  —  kilo.;  grain  =  gr. ;  gramme  =  gm.;  kilo 
gramme  —  k.;  kilogramme  metre  =  km. 

i  British  ton  —  2240  Ibs. 

=  1016  k. 
i  U.  S.  ton      =  2000  Ibs. 

=  907-143  k- 


I  in. 

—  2.54  cm. 

i  cm. 

=  -39370II  in. 

I  ft. 

=  30.4709  cm. 

i  m. 

—  3.280843  ft. 

i  mile 

=  1.6093  kilo. 

I  kilo. 

=  .62137  mile. 

i  knot 

=  i  naut.  mile  per  hr. 

=r  6080  ft.  (av.)  per  hr. 

i  sq.  in. 

=  6.4516  cm.2 

i  cm.9 

=  .155  sq.  in. 

i  sq.  ft. 

=  929.03  cm.1 

i  m.2 

=  10.7639  sq.  ft. 

i  sq.  yd. 

—  .  836126  m.* 

I  acre 

=  43.56osq.  ft. 

=  .40468  hectare. 

i  hectare 

=  10,000  m.f 

=  loo  ares. 

—  2.4711  acres. 

I  sq.  mile 

=  640  acres. 

—  2.59  sq.  kilo. 

=  259  hectares. 

I  sq.  kilo. 

==  100,000  m.2 

=  24.711  acres. 

I  Ib. 

—  16  oz.  =  7000  gr. 

=  -4535924  k. 

=  453-  5924  gm. 

=  445,000  dynes. 

Ik. 

=  2  204622  Ibs. 

=  981,000  dynes. 

i  Fr.  tonne 


=  loco  k. 

=  .9842  British  ton. 

=  2204.622  Ibs. 


I  cu.  in.  of  water  at  4°  C.  =  252.89  gr. 
i  cm.'  "  =  i  gm. 

i  cu.  ft.        "  "        =  62.43  Ibs. 

i  litre  "  "         —  i  k. 

i  imp.  gal.  at  62°  F.          =  10  Ibs. 
I  cu.  ft.  of  water  at  62°  F.  =  62.3  Ibs. 
i  cu.  ft.  of  air  at  o°  C. 


and  i  atm. 


=  .0807  Ib. 


I  cu.  ft.  of  hydrogen  at  ) 

o°  C.  and  .  atm.          {"""SHS. 
i   litre   of  air  at  o°  C. 

and  i  atm. 


Water   compresses 


of    its 


bulk  under  a  change  of  pressure  of 
i  atm.,  or  about  -^th  of  its  vol.  un- 
der a  pressure  of  2  tons  (of  2240  Ibs.) 
per  sq.  in. 

i  Ib,  per  sq.  in.   =  .0703  k.  per  cm.2 

I  k.  per  cm.2        =  .0703  lb.persq.in. 

i  Ib.  per  sq.  ft.    =  4.8826  k.  per  m.2 

=  479-dynespercm.2 

xvii 


xviii 

USEFUL   CONSTANTS. 

No.  of  Ibs.  per  ) 
sq.  in.             j" 

=  14.223  k.  per  cm. 

i  standard  atm.  1                        ins      of 
ofi4.7lbs.per^|       me 

(  .4907     ins.      of 
~  I      mercury. 

sq.  in.                 J 
=  760  mm. 
i  metric  atm.  of  1 

j  ins.  of  mercury 

~     \               -T-    2.0378. 

-,       (  28.96    ins.    of 
14.  223  Ibs.  per  }-  =  3 
sn    in                '          <       mercury. 

No.  of  k  per  m.s 

_  j  4.8826  Ibs.    per 

sq.  in.  -              J 
i  erg                         =  i  dyne  X  i  cm. 

~  (      sq.  ft. 

i  gm.-cm.                 =  981  ergs. 

i   in.    of   mer-  ) 
cury  at  o°  C.  ) 

=  .  034534k.  per  cm.2 

i  ft.-lb.                      =  .13825  km. 
=  1-3562  Xio7ergs. 

I  mm.  mercury  j 

.       o  y» 

=  .0013596  per  cm.2 

i  km.                        =  7.233  ft.-ibs. 

at  o   C.            ) 

=  9.81  X  io7  ergs. 

No.  of  ft.-lb.           =7.2178  km.. 

i  cu.  in. 

=  16.387  cm.s 

=  777  B.  T.  U. 

i  cm.3 

I  CU.   ft. 

=  .061  cu.  in. 
=  .028317  m.J 

_  (  1399    Ibs.   de- 
(      gree  C. 

=  28.317  litres. 

i  B.  T.  U.                =  1058  joules. 

i  m8. 

=  35.3U8   cu.   ft. 

=  1058  X  io1  ergs. 

I  litre 

=  1000  cm.3 

i  k.  degree  C.         =  4200  joules. 

=  1.7598  pints. 

=  4200  X  io7  ergs. 

=  .22  imp.  gal. 

i  calorie                    =  i  k.  raised  i°  C. 

i  imp.  gal. 

=  .1605  cu.  ft. 

=  426.9  km. 

=  277.27  cu.  ins. 

=  3080.9  ft.  -Ibs. 

=  4.545963  litres. 

i  watt                        =  i  joule  per  sec. 

i  U.  S.  gal. 

=  231  cu.  ins. 

f  work  done  by 

f,. 

=  .83254  imp.  gal. 
(  981     cm.     per 
~  (  sec.  per  sec. 

J       a  current  of 
1       i  amp.  at  i 
[      volt. 

_  j  32.2     ft.     per 

} 

i  horse-power        =  550  Ibs.  per  sec. 

sec.  per  sec. 

(  746  X  io9  ergs 

g  at  Greenwich 

=  32.19078  ft. 

=  •! 

}      per  sec. 

=  981.17  cm. 

=  746  watts. 

g  at  London 

=  32.182  ft. 

_  j  i.oi  forcesrde- 

=  980  9  cm. 

~  \      cheval. 

g  at  Manchester 

=  32.196  ft. 
=  981.34  cm. 

iforce-de-cheval=  (-9863       horse- 
(      power. 

g  at  the  equator 

=  32.088  ft. 

=  736  watts. 

g  at  Baltimore 

=J978.O4  cm. 
=  32.152  ft. 

_  j  545  ft.  -Ibs.  per 
~~  /      sec. 

=  980  cm. 

=  75  km.  per  sec. 

g  at  Montreal 

=  32.1765  ft. 

i  radian                   =57.296   degrees. 

=  980.73  cm. 

To  convert  common    into  hyper- 

The   inertia    or 
mass  of  a  body 

f  its  wt.  in  Ibs. 
I  =  \      at    London 
'        [     -4-32.2. 

bolic  and  hyperbolic  into    common 
logarithms,  multiply  the  former  by 
2.3025  and  the  latter  by  .43429. 

HYDRAULICS, 


CHAPTER    I. 

GENERAL  PRINCIPLES.     FLOW  THROUGH  ORIFICES,  OVER; 

WEIRS,    ETC. 

I.  Fluid  Motion. — The  term  "  hydraulics,"  as  its  deriva- 
tion (vdoop,  water;  <*uAo.c,  a  tube  or  pipe)  indicates,  was 
primarily  applied  to  the  conveyance  of  water  in  a  tube  or  pipe, 
but  its  meaning  now  embraces  the  experimental  theory  of  the 
motion  of  fluids. 

The  motion  of  a  fluid  is  said  to  be  steady  or  permanent 
when  the  molecules  successively  arriving  at  any  given  point 
are  animated  with  the  same  velocity,  are  subjected  to  the  same 
pressure,  and  are  the  same  in  density.  As  soon  as  the  motion 
of  a  stream  becomes  steady  a  permanent  regime  is  said  to  be 
established,  and  hydraulic  investigations  are  usually  made  on 
the  hypothesis  of  a  permanent  regime.  With  such  an  hypothe- 
sis, any  portion  of  the  fluid  mass,  which  leaves  a  given  region, 
is  replaced  by  a  like  portion  under  conditions  which  are  identi- 
cally the  same. 

The  terms  "steady  motion"  and  "permanent  regime" 
are  often  considered  to  be  synonymous. 


a  FLUID  MOTION. 

The  general  problem  of  flow  is  the  determination  of  the 
relation  which  exists  at  any  point  between  the  density,  pres- 
sure, and  velocity  of  the  molecules  which  successively  pass  that 
point. 

The  actual  motion  of  a  fluid  is  exceedingly  complex,  and, 
In  order  to  simplify  the  investigations,  various  assumptions  are 
made  as  to  the  nature  of  the  flow. 

2.  (a)  Stream-line  Motion. — The  molecules    may  be  re- 
garded as  flowing  along  definite  paths,    and   a    succession  of 
such  molecules  as  forming  a  continuous  fluid  rope,   which  is 
termed  an  elementary  stream  or   a   fluid    filament  ;   or,    if  the 
motion   is  steady,   and  the    paths  therefore  fixed   in    space,  is 
termed  a  stream-line. 

Experiment  shows  that  the  velocity  of  flow  in  any  cross- 
section  varies  from  point  to  point,  and  it  is  often  assumed  that 
the  section  is  made  up  of  an  infinite  number  of  indefinitely 
small  areas,  each  area  being  the  section  of  a  fluid  filament. 

(b)  Motion  in  Plane  Layers. — In  this  motion  it  is  assumed 
that  the  molecules,  which  at  any  given  moment  are  found  in  a 
plane  layer,  will  remain  in  a  plane  layer  after  they  have  moved 
into  any  new  position. 

(c)  Laminar  Motion. — On  this   hypothesis   the   stream    is 
supposed  to  consist  of  an  infinite  number  of  indefinitely  thin 
layers.      The   variation    in    velocity  from  point  to    point   of  a 
cross-section   may  then  be  allowed  for,  by  giving  the  several 
layers  different  velocities  based  upon  the  law  of  fluid  resistance 
between  consecutive  layers. 

3.  Density;  Compressibility;  Head;  Continuity. 

The  freezing-point  of  pure  water  is  32°  F.  or  o°  C. 
"  boiling-  "  "  "  "  "  212°  F.  or  100°  C. 
"  max.  density  "  "  "  44  39°.  i  F.  or  4°  C. 
-"  standard  mean  temperature  "  62°  F.  or  i6°.66  C. 

The  comparative  densities  and  also  the  comparative  vol- 
umes are  the  same  at  32°  F.  and  46°  F. 


FLUID  DENSITY.  3 

The  bulk  of  fresh  snow  is  12  times  the  bulk  of  the  equiva- 
lent water. 

i  cu.  ft.  of  fresh  snow  weighs  5.2  Ibs.  and  its  s.  g.  is  .0833. 

I    cu.  it.  of  ice  at   32°  F.  weighs   57  J  Ibs.  and  its  s.  g.  is 

.922. 

i   cu.  ft.  of  average  sea-water  at  62°  F.  weighs  64  Ibs.  and 
Its  average  s.  g.  is  1.028. 

i  cu.  ft.  of  pure  water  at  32°  F.  weighs  62.418  Ibs. 

*'      "  39°.  i  F.     "       62.425    " 
44          44      44  52°. 3  F.     "       62.400   " 

«  44  44  44  44     520      p  44  62.   355         *' 

44  44      2I20      p  44  59.640        " 

6.2355  gallons  or 


contains  ,  ,,  . 

6.2328  imperial  gallons. 

i  cu.  yd.        "          "  "        168.36  gallons. 

i  cu.  metre    "  "  "        220.09        " 

The  vol.  of  i  Ib.  of  pure  water  at  32°  F.     '  is  .016021  cu.  ft. 

"              "         l<            39°.  i  F.    "  .016019  i4 

52°.  3  F.    <4  .016  « 

62°  F.       "  .016037  " 

j20     p                    t<     .01677  '* 

The  vol.  of  i  ton         "         "             52°.  3  F.    "35.9  <4 

44       4<            4t       sea-water  at           62°  F.        "35  " 

i  tonne  of  pure  water  at  39°.  i  F.    "35.3156  " 

-0353  4< 


44       4' 


i  gallon  of  pure  water  at  62°  F.  weighs  10  Ibs.  and  its  vol. 
=  277.123  cu.  ins.  =  .16037  cu-  ft- 

i  imperial  gallon  of  pure  water  at  62°  F.  weighs  10.00545 
Ibs.  and  its  vol.  =  277.274  cu.  ins. 


FLUID  DENSITY. 


|P 

!  ! 

i       & 

III    I 

|J 

CO                   "1 

M                           M 

•-.a 

CO                  M 

^'           <£ 

MOO                           O         M 

11 

M                    M 

0               0 

M                           M    ' 

0                     0 

MMM                                   O            t^ 

OOO                            O          vn 

nparative 
Density. 

O         CO  O          -1-  M 
M            OO            M    r>i 

O         !->•  u->        co  O 
co        co  r-»        r^  r^- 

M             Q     h^     d             U")    ON  1O 

co         m  o*    O       \O    c^  O 
vO        O  *O   w~i         tr>  10  10 

**•  ''I-  rfr  CO  CO           MMM 

o 

U 

-T3  <fi 

0 

0                                  Tf 

M                                r>.                   vn  co       co 

0.2  Jj 

M                         CO 

O                                   Tf 

-•^b/D 

M      M      C4      CO    CO    Tf    IT) 

o  o  r^co  o  o  o  M  w 

M    CO  *^"  u^  \O    f^*  1^  CO    O  O    u~>  CO  O  CO 

^3° 

MM 

d|| 

OO    O  O    **?           M 

CO           O     ~f  M                  CO  O 

~t  M           CO  O            Tj-  M 
O    M    co  10  O  co    O    O    M    *•?•  O    O    M    O 

ta 

.S   0   c' 

5                 ?                         MM 

co  M              r^ 

O  O             co 

M            CO                          CO           O                    M 

4-1     [A      O 

O          O               O   O 

O   O               0 

88           8     8        § 

si- 

O    '     O                0    O 

O   O               O 

MM                       M 

O          O                     O          O                0 

M                 M                                      M                 M 

ll 

co         M               \r>  \r> 

M  CO                   O 

MM                    O 

8-1-                          M            vn                   -J- 
O                              !~»            "">                      'I' 

•-'.5 

"**•<$•              •**•** 

*t           CO                          CO          CO                  CO 

•c  3 

MM                    MM 

MM                    M 

MM                           MM                    M 

'«i 

00               00 

O  O               O 

O         O                       O          O                O 

•^"o 

u 

oo  r-       co  M   O 

•     -f-TfMCO           M           ^--l-          *^-M 

'w    ^ 

O    M          OOO 

oo  M     .        r^  o  M  co 

M-or^-M         r^        OM         coco 

ct   ~ 

CO    O         OOO 

ooo    •        co  co  r^o 

in     •     -T  N    M            O         COO           ~f  M 

II 

O  O         O  O-  O 
O  O         OOO 

O-OOO       co        coco         coco 

OM 

w 

U 

ilS 

S           5 

0   ^ 
O  .r-» 

rc         oo                      vn        O                co 
M          r^                     u->        o                co 

I'lftx- 

o  M  M  M  co  TJ-  ;^ 

u~)  \o  i^.  r>»  r^»  co   o  O   M 

M.M   M   coTj-i/->u->voo   rococo   OO 

U 

r'C  <r> 

• 

co         O    ^t"  *~^  - 

CO  O                    -^-  M           CO 

tOO            "3-  M                  CO           O    ~t           M 

|ji  & 

Cl    ro  10  u->  r»  O  O 

M      N      T^-    U-.O    O    CO      O      M 

M    covouor^OO    O    M    M    -txnoco 

[2 

FLUID   COMPRESSIBILITY.  5 

The  temperatures  in  this  table  may  be  taken  as  abscissae,  and 
trm  corresponding  values  in  the  three  remaining  columns  as 
ordinates.      Curves  of  comparative  density,   weight  per  cubic 
oot,  and  weight  per  gallon  are  thus  obtained,  and  the  values 
orresponding  to  any  specified  temperature  can  be  easily  and 
/ery  accurately  determined  from  these  curves  by  direct  meas- 
urement. 

The  weights  per  cubic  foot  in  the  table  have  been  calcu- 
lated by  means  of  Rankine's  approximate  formula, 

w  1000 T 

62.425  "      T*  -(-  250,000  ' 

•w  being  the  weight  per  cubic  foot  corresponding  to  the 
-absolute  temperature  T,  i.e.,  461°  -)-  ordinary  temperature. 

The  specific  weights  obtained  by  this  rule  for  the  lower 
temperatures  are  very  exact,  but  for  the  higher  temperatures 
they  become  too  large.  Thus  the  rule  gives  59.76  Ibs.  as  the 
weight  of  a  cubic  foot  of  pure  water  at  212°  F.,  while  actual 
measurement  makes  the  weight  59.64  Ibs. 

The  comparative  densities  between  o°  C.  and  40°  C.  are 
the  values  obtained  by  Chappuis. 

Compressibility. — Fluids  are  sensibly  compressible  under 
heavy  pressures,  and  the  compression  is  proportional  to  the 
pressure  up  to  about  1000  Ibs.  (68  atmospheres)  per  sq.  inch. 
Grassi's  experiments  indicate  that  the  compressibility  of  water 
diminishes  as  the  temperature  increases.  Water  compresses 

about  47i  millionths  (i.e.,    ^^  =  ^^,   nearly)    of  its 

bulk  for  each  atmosphere.  This  is  equivalent,  approxi- 
mately, to  a  reduction  of  y1^  in  the  bulk  under  a  pressure  of 
2  tons  per  sq.  inch. 

If  a  volume  F  of  a  fluid  is  compressed  by  an  amount  A  V 
under  an  increase  Ap  of  the  pressure,  then  the  amouut  of  com- 
pression per  unit  of  vol.  is 

AV 

-FT-  and  is  called  the  cubical  compression.      The  ratio  of  the 


FLUID   COMPRESSIBILITY. 
TABLE  OF  ELASTICITY  OF  VOLUME  OF  LIQUIDS. 

(Reduced  from  Grassi's  results.) 


Liquid. 

Elasticity  oi  Voiume. 

Temperalure. 

Mercury  .  .. 
Water  . 

717,000,000 
j  42,000,000 

o°      C. 
o°      C. 

Sea-  water  .  . 
Ether  

Alcohol.... 
Oil  

}  45,900,000 
52,900,000 
(  16,280,000 
I  15,000,000 
j  25.470,000 
I  23,380,000 
44,000  ooo 

18°      C. 

o°      C. 
14°      C. 
7-3°C. 
13.1°  C. 

N.  B.  —  The  value  for  mercury  is  probably  erroneous. 

increment  of  pressure  to  the  cubical  expansion,  viz., 


is   termed  the   elasticity  of  volume.      This  is  sensibly  constant 
within    wide  limits,   and  is  generally  denoted    by    the  letters 

Dor  K. 

The  vertical  distance  between  the  free  surface  of  a  mass  of 
water  and  any  datum  plane  is  called  the  head  with  respect  to 
that  plane.  If  the  water  extends  down  to  the  level  of  the 
plane,  a  pressure  p  is  produced  at  that  level,  and  the  value  of 
/,  so  long  as  the  water  is  at  rest,  is  given  by  the  equation 


W  W 

w  being  the  specific  weight  of  the  water  and  pQ  the  pressure  at 
the  free  surface.  Thus  the  pressure  may  be  measured  in  terms 
of  the  head,  and  hence  the  expression  "  head  due  to  pressure  " 
or  "  pressure  head." 

A  column  of  water  at  62°  F.  and  2.3093  ft.  in  height 
exerts  a  pressure  of  I  Ib.  per  sq.  inch. 

A  column  of  water  at  62°  F.  and  33.947  ft.  or  10.347 
metres  in  height  exerts  a  pressure  of  14.7  Ibs.  per  sq.  in.,  or 
one  atmosphere. 

A  column  of  water  at  62°  F.  and  I  ft.  in  height  exerts  a. 
pressure  of  .433  Ibs.  per  sq.  inch. 


HEAD  OF  WATER.  ^ 

Head.  —  A  head  of  water  is  a  source  of  energy.      A  volume 
o£  water  descending  from  an  upper  to  a  lower  level  may  be 
employed  to  drive  a  machine,  which  receives  energy  from  the 
water  and  again   utilizes  it  in  overcoming  the    resistances  o 
other  machines  doing  useful  work. 

Let  Q  cu.  ft.  of  water  per  second  fall  through  a  vertica. 
distance  of  //  ft.  Then  the  total  power  of  the  fall  =  wQlt 

w  Qh 
ft.-lbs.,  =        -  h.p.  ,  w  being  the  weight  of  the  water  in  pounds- 

per  cubic  foot. 

Let  K  be  the  proportion  of  the  total  power  which  is. 
absorbed  in  overcoming  frictional  and  other  resistances.  Then 
the  effective  power  of  the  fall  =  wQ/i(i  —  K),  and  the  effi- 
ciency is  I  —  K. 

Continuity.  —  Imagine  abounding  surface  enclosing  a  space 
of  invariable  volume  in  the  midst  of  a  moving  mass  of  fluid. 
The  principle  of  continuity  affirms  that,  in  any  interval  of  time, 
the  flow  into  the  space  must  be  equal  to  the  outflow  during  the 
same  interval.  Giving  the  inflow  a  positive  and  the  outflow  a 
negative  sign,  the  principle  may  be  expressed  symbolically  by 


The  continuity  of  a  mass  of  water  will  be  preserved  so  long 
as  the  pressure  exceeds  the  tension  of  the  air  held  in  solution. 
It  is  on  account  of  the  pressure  of  this  air  that  pumps  cannot 
draw  water  to  the  full  height  of  the  water-barometer,  or  about 
34ft. 

Generally  speaking,  the  pressure  at  every  point  of  a  con- 
tinuous fluid  must  be  positive.  A  negative  pressure  is  equiva- 
lent to  a  tension  which  will  tend  to  break  up  the  continuity 
presupposed  by  the  formulae.  Should  negative  pressures  result 
from  the  calculations,  the  inference  would  be  that  the  latter 
are  based  upon  insufficient  hypotheses. 

The  pressure  in  water  flowing  through  the  air  cannot  at  any 
point  fall  below  the  atmospheric  pressure.  There  are  cases,, 


8  BERNOULLI'S    THEOREM. 

however,  as  when  water  flows  through  a  closed  pipe  (Art.  6, 
Chap.  II),  in  which  the  pressure  may  fall  below  this  limit  and 
become  almost  nil.  In  this  case  there  is  a  danger  of  the  air 
held  in  solution  being  set  free,  thus  tending  to  interrupt  the 
continuity  of  the  flow,  which  may  even  be  wholly  stopped  if 
the  air  is  present  in  sufficient  volume. 

Consider  a  length  of  a  canal  or  stream  bounded  by  two 
normal  sections  of  areas  Alt  A2,  and  let  7',,  ?>.,  be  the  mean 
normal  velocities  of  flow  across  these  sections.  Then,  by  the 
principle  of  continuity, 


and  the  velocities  are  inversely  as  the  sectional  areas. 

Again,  assume  that  a  moving  mass  of  fluid  consists  of  an 
infinite  number  of  stream-lines,  and  consider  a  portion  of  the 
mass  bounded  by  stream  -lines  and  by  two  planes  of  areas  Ax  , 
A2  at  right  angles  to  the  direction  of  flow.  If  i\  ,  ?'.,  are  the 
mean  velocities  of  flow  across  the  planes, 

vlAl  =  Q  =  i'2A  if  the  fluid  is  incompressible. 

Assuming  that  the  fluid  is  compressible,  and  that  the  mean 
specific  weights  at  the  two  planes  are  wl  and  w.,  ,  then  the 
weight  of  fluid  flowing  across  A.Y  is  equal  to  the  weight  which 
flows  across  A2  ,  since  the  weight  of  fluid  between  the  two  planes 
remains  constant.  Hence 


4.  Bernouilli's  Theorem.  —  This  theorem  is  based  on  the 
following  assumptions: 

(1)  That  the  fluid  mass  under  consideration  is  a  steadily 
moving  stream  made  up  of  an  infinite  number  of  stream-lines 
whose  paths  in  space  are  necessarily  fixed. 

(2)  That  the  velocities  of  consecutive  stream-lines  are  not 
widely  different,  so  that  viscosity,  or  the  frictional  resistance 
between  the  stream-lines,  is  sufficiently  small  to  be  disregarded. 


BERNOULLI'S   THEOREM. 


(3)  That  the  fluid  is  incompressible,  so  that  there  can  be 
no. internal  work  due  to  a  change  of  volume. 

In  any  given  stream-line  let  a  portion  AB,  Fig.  I,  of  the 
fluid  move  into  the  position  A'B  in  t  seconds. 


FIG.  i 


Let  al  ,  pi  ,  i\  ,  zl  be  the  normal  sectional  area,  the  intensity 
of  the  pressure,  the  velocity  of  flow,  and  the  elevation  above 
a  datum  plane  zz  of  the  fluid  at  A.  Let  a2,  /2,  z/2,  #2  denote 
similar  quantities  at  B. 

Since  the  internal  work  is  nil,  the  work  done  by  external 
forces  must  be  equivalent  to  the  change  of  kinetic  energy. 

(  the  work  done  by  gravity 
(  -j-  the  work  done  by  pressure. 
But  when  the  fluid  AB  passes  into  the  position  A'B',  the 
work  done  by  gravity  is  equivalent  to  the  work  done  in  the 
transference  of  the    portion  BB'  ',   and  therefore,   t  being    the 
time, 

the  work  done  by  gravity  =  wal  .  AA'.  zl  —  Z£/#2  .  BB'  .  zz 

~  wa  .  it  .       —  wa  .  vt  .  z 


Now  the  external  work  = 


since  A  A'  =  it,  BB'  = 


and 


\  =  Q  =  a2vr 


Again,  the  ivork  done  by  the  pressures  on  the  ends  A  and  B 


The   work   done  by  the   pressure   on   the   surface    of  the 


io  BERNOUILLrS    THEOREM. 

stream-line  between  A   and  B  is  nil,  since  the  pressure  is  at 

every  point  normal  to  the  direction  of  motion, 
The  change  of  kinetic  energy 

=  kinetic  energy  of  A'  B'  —  kinetic  energy  of  AB 
=  kinetic  energy  of  BB'  —  kinetic  energy  of  A  A', 

since  the  motion  is  steady,  and  there  is  therefore  no  change  in 

the  kinetic  energy  of  the  intermediate  portion  A'  B.      Thus 

the  change  of  kinetic  energy  =  -  aJBB'  —  ---  a^AA'- 

W  7>2         W  V,2 

*— 


w  ^  (v?       7A2\ 
=  -Ot  I-2-  —  -^-  . 

g  "     \  2  2  / 


Hence,    equating   the   external   work   and   the    change   of 
kinetic  energy, 


which  may  be  written  in  the  form 


w  v*  w  v  2 

-       =  ^2  +  A+-.     •     •    (i). 


or 


But  ^4  and  .#  are  arbitrarily  chosen  points,  and  therefore, 
at  any  point  of  a  stream-line,  the  motion  being  steady  and  the 
viscosity  nil,  the  gradual  interchange  of  the  energies,  due  to 
head,  pressure,  and  velocity,  is  expressed  by  the  equation 

W  V2 
wz  +  P  H  ---  L  =  WH,  a  constant  ;   .     .     .     (3) 

o 

or 

Z  +  w  +  ^  =  H>  a  Constant5        ...      (4) 

Z  being  the  elevation  of  the  point  above  the  datum  plane,  p  the 
pressure  at  the  point,  w  the  specific  weight,  and  v  the  velocity 
of  flow.  This  is  Bernouilli's  theorem. 


BERNOULLI'S   THEOREM.  II 

Thus  the  total  constant  energy  of  wH  ft.-lbs.  per  cubic  foot 
of  fluid,  or  H  ft.-lbs.  per  pound  of  fluid,  is  distributed  uniformly 
along  a  stream-line,  wH  being  made  up  of  wz  ft.-lbs.  due  ta 


head,   p  ft.-lbs.  due  to  pressure,  --  ft.-lbs.  due  to  velocity, 

p 

and  H  being  made  up  of  z  ft.-lbs.  due  to  head,  —  ft.-lbs.  due 

w 

V2 
to  pressure,  and  —  ft.-lbs.  due  to  velocity. 

Hence  the  total  energy  is  made  up  of  three  elements,  and 
each  element  may  be  utilized  by  a  specially  designed  motor. 
The  now  almost  obsolete  overshot-wheel  is  driven  by  the 
weight  of  the  water  filling  the  buckets  on  one  side  and  descend- 
ing from  a  higher  to  a  lower  level.  In  the  breast-wheel  and 
certain  turbines,  the  energies,  due  both  to  the  weight  (wz)  and 

lwv\ 
to  the  velocity  (  -  1,  are  transformed   into  useful  work.      The 

/  wv*  \ 
rotation  of  impulse-wheels  is  due  to  the  kinetic  energy  (  -  j: 

of  a  jet  of  water  issuing  from  a  nozzle  and  impinging  upon 
curved  buckets.  Finally,  the  piston  of  the  hydraulic  engine  is. 
actuated  by  water  admitted  into  the  cylinder  from  a  closed 
pipe  in  which  the  water  under  pressure  moves  with  a  low* 
velocity. 

Assuming  that 

(a)  the  motion  is  steady, 

(<£)  the  frictional  resistance  may  be  disregarded, 

(c)  the  fluid  is  incompressible, 

Bernoulli's  theorem  may  be  applied  to  currents  of  finite  size 
at  any  normal  section,  if  the  stream-lines  across  that  section 
are  sensibly  rectilinear  and  parallel.  Tr^ere  is  then  no  interior 
work  due  to  a  change  of  volume,  and  the  distribution  of  the 
pressure  in  the  section  under  consideration  will  be  the  same  as 
if  the  fluid  were  at  rest,  that  is,  in  accordance  with  the  hydro- 
static law.  This  is  also  true  whether  the  flow  takes  place 


12  APPLICATIONS   OF  BERNOULLI'S    THEOREM. 

under  atmospheric  pressure  only,  or  whether  the  fluid  is  wholly 
or  partially  confined  by  solid  boundaries,  as  in  pipes  and 
canals,  or  whether  the  flow  is  through  another  medium  already 
occupied  by  a  volume  of  the  fluid  at  rest  or  moving  steadily  in 
a  parallel  direction.  In  the  last  case  there  must  necessarily 
be  a  lateral  connection  between  the  two  fluids,  but  the  pressure 
over  the  section  must  follow  the  hydrostatic  law  throughout  the 
separate  fluids,  and  there  can  be  no  sudden  change  of  pressure 
at  the  surface  of  separation,  as  this  would  lead  to  an  interrup- 
tion of  the  continuity. 

The  hypotheses,  however,  upon  which  these  results  are 
based,  are  never  exactly  realized  in  actual  experience,  and  the 
results  can  only  be  regarded  as  tentative.  Further,  they  can 
only  apply  to  an  indefinitely  short  length  of  the  current,  as  the 
viscosity,  which  is  proportional  to  the  surface  of  contact,  would 
otherwise  become  too  great  to  be  disregarded. 

5.  Applications  of  Bernouilli's  Theorem.  —  If  a  glass  tube, 
open  at  both  ends,  and  called  a  piezometer  (nie^eiv,  to  press; 

jjerpov,  a  measure)  is  inserted 
vertically  in  the  current,  Fig.  2, 
at  a  point  N,  z  ft.  above  the  point 
0  in  the  datum  line,  the  water 
will  rise  in  the  tube  to  a  height 
MN  dependent  upon  the  pressure 
at  N.  The  effect  of  the  eddy 
motion,  produced  at  TV  by  obstruct- 
i  ing  the  stream-lines,  may  be  dimin- 

ished by  making  this  end   of  the 

_  j,  _  tube    parallel   to   the   direction    of 

flow.      Disregarding  the    effect    of 
the  eddies  and  taking  /  and  /0  to 

be  the  intensities  of  the  pressure  at  N  and  of  the  atmospheric 
pressure,  respectively,  then, 


w  w 


APPLICATIONS  OF  BERNOULLI'S   THEOREM. 
and  therefore 


- 

w 


tv 


=  ON  -\-MN-\-     ~ 


w 


The  locus  of  all  such  points  as  J/is  often  designated  "the 
line  of  hydraulic  gradient,"  or  the  "virtual  slope,"  terms  also 
used  when  friction  is  taken  into  account. 

Let  the  two  piezometers  AB,  CD,  Fig.  3,  be  inserted  in 
the  current  at  any  two  points  B  and  D,  z^  ft.  and  32  ft. 
respectively  above  the  points  E  and  F'm  the  datum  line. 


FIG.  3. 


/l  be  the  intensity  of  the  pressure  at  B  in  pounds  per 
square  foot,  /2  that  at  D,  and  let  the  water  rise  in  these  tubes 
to  the  heights  BA,  DC.  Then, 


A. 

w 


— ,'    and 


14  APPLICATIONS   OF  BERNOULLI'S   THEOREM. 

and  therefore 


(6) 


the  line  AG  being  parallel  to  the  datum  line. 

Thus,  (z.  +  -1)  —  (V,  +  -2)  is  equal  to  the  fall  of  the  free 
\  1    '    wl        \  2       iv  I 

surface  level  between  the  points  B  and  D. 

Let  v^  ,  7>2  be  the  velocities  of  floW  at  B  and  D.      Then,  by 
Bernoulli's  theorem, 


and  therefore  the  fall  of  free  surface  level  between  B  and  D 


Equation  (7)  may  also  be  written  in  the  form 

+  ~)  -  (*•  +  ~)  =  ^  +  CG*     '      (8) 


so  that  the  velocity  at  D  is  equal  to  that  acquired  by  a.body 
with  an  initial  velocity  i\  falling-  freely  through  the  vertical 
distance  CG. 

Froude  illustrated  Bernoulli's  theorem  experimentally  by 
means  of  a  tube  of  varying  section,  Fig.  4,  conveying  a  current 
between  two  cisterns.  The  pressure  at  different  points  along 
the  tube  is  measured  by  piezometers,  and  it  is  found  that  the 
water  stands  higher  and  the  pressure  is  therefore  greater, 
where  the  cross-section  is  larger  and  the  velocity  consequently 
less.  Reynolds  illustrates  the  principle,  that  the  pressure  in  a 
frictionless  pipe  of  varying  section  increases  and  diminishes 
with  the  section,  by  forcing  water  at  a  high  velocity  through  a 
:J-in.  pipe  drawn  down  in  the  middle  to  a  bore  of  .05  inch.  At 


APPLICATIONS   OF  BERNOULLI'S    THEOREM.  15 

this  point  the  pressure  is  so  much  diminished,  that  the  water 
hisses  and  boils.  To  the  same  cause  is  due  the  hissing  sound 
heard  in  water-injectors  and  in  partially  opened  valves.  If  the 
section  of  the  throat  at  A  is  such,  that  the  velocity  is  that 
acquired  by  a  body  falling  freely  through  the  vertical  distance 


FIG.  4. 

h  between  A  and  the  surface  level  of  the  water  in  the  cistern, 
and  if  p  be  the  pressure  at  A ,  and  z  the  elevation  of  A  above 
•datum,  then,  neglecting  friction, 

p         7>2  P 

z  +  —  +     -  =  H  =  s  +  //  +  /0. 

1  '  W  2g  W 

But  ?'2  =  2g/i,  and  therefore  p  =  /0 ,  so  that  the  pressure 
at  A  is  that  due  to  atmospheric  pressure  only.  Thus,  a 
portion  of  the  pipe  in  the  neighborhood  of  A  may  be  removed, 
as  in  the  throat  of  the  injector. 

Again,  let  the  cross-section  in  the  throat  at  B  be  less  than 
that  at  A.  The  pressure  at  B  will  be  less  than  the  atmos- 
pheric pressure,  and  a  column  of  water  will  be  lifted  up  in  the 
urved  piezometer  to  a  height  //'. 

Let  a^ ,  slt  /! ,  i\  be  the  sectional  area,  elevation  above 
datum,  pressure,  and  velocity  at  B. 

Let  a2 ,  #2 ,  /2 ,  7'2  be  similar  symbols  at  E. 


16  APPLICATIONS  OF  BERNOUILU'S   THEOREM. 

Then 


P 

Put  H2  =  ^2  -f-  ~    >  the   height  above   datum  to  which   the 
water  is  observed  to  rise  in  the  piezometer  inserted  at  £t  and 

also  let  H.  =  s.  +  ~°  -  //'.      Then 

1    zc; 


*  i        2g  2g       a. 

since  a.2v2  =  a^\  ,  «,  being  the  sectional  area  at  E.      Therefore 


an  equation  giving  the  theoretical  velocity  of  flow  at  the 
throat  B.  Hence  the  theoretical  quantity  of  flow  across  the 
section  at  B  is 


\'a*  -  a* 


V2g(H2  -  //J.       .     .     (10) 


This  is  the  principle  of  the  aspirator  and  also  of  the  Venturi 
water-meter,  which,  as  now  used,  is  said  to  be  correct  to 
within  £  per  cent. 

The  actual  quantity  of  flow  is  found  by  multiplying  equa- 
tion (10)  by  a  coefficient  C,  whose  value  is  to  be  determined 
by  experiment  and  may  be  taken  to  be  approximately  unity. 

If  the  pressure  at  E  is  positive,  then  H2  is  merely  the 
height  to  which  the  water  is  observed  to  rise  in  an  ordinary 
piezometer  inserted  at  E. 

Again,  Froude  also  points  out  that  when  any  number  of 
combinations  of  enlargements  and  contractions  occur  in  a  pipe, 
the  pressures  on  the  converging  and  diverging  portions  of  the 


EXAMPLES.  if 

pipe  will  balance  each  other  if  the  sectional  areas  and  direc- 
tions* of  the  ends  are  the  same. 

Ex.  i.  One  cubic  foot  of  water  per  second  flows  steadily  through  a 
frictionless  pipe.  At  a  point  A,  100  ft.  above  datum,  the  sectional  area 
•of  the  pipe  is  .125  sq.  ft.,  and  the  pressure  is  2500  Ibs.  per  sq.  ft.  Find 
the  total  energy  at  A  per  cubic  foot  of  water.  At  a  point  B  in  the  datum 
line,  the  pressure  is  1250  Ibs.  per  sq.  ft.  and  the  sectional  area  .0625 
3q.  ft.  Find  the  loss  of  energy  per  cubic  foot  of  water  between  A  and  B. 

The  velocity  of  flow  at  A  =  -  =  8  ft.  per  sec. 
The  total  energy  at  A  per  cubic  foot  of  water 


The  velocity  of  flow  at  B  =  —  >—  =  16  ft.  per  sec. 

.0025 

The  total  energy  at  B  per  cubic  foot  of  water 

1250         i62 
=  °  +  -65T  +  -64-  ==  24  ft-lbs' 

Hence,  the  loss  of  energy  between  A  and  B  per  cubic  foot  of  water 
=  141  —  24  =  117  ft.  -Ibs. 

Ex.  2.  A  horizontal  frictionless  pipe,  in  which  the  pressure  is  100  Ibs. 
per  square  inch,  gradually  contracts  to  a  throat  of  one  tenth  of  the 
diameter  and  then  again  gradually  enlarges  to  a  pipe  of  uniform  diame- 
ter. What  will  be  the  maximum  velocity  of  flow  at  the  throat  ? 

The  velocity  at  the  throat  will  be  greatest  when  the  pressure  there 

is  nil.     Hence,   if  v  is  the  throat  velocity  and  therefore  —  the  pipe 

velocity, 

TOO  x  144        i  /  v  \*_  v1 

62^        *  64\mo/  ~        f  64' 
and  v  =  121.437  ft.  per  sec. 

6.  Rotation  of  a  Fluid.  —  In  any  stream-line  moving  freely 
in  space,  let  A  BCD  be  an  element  of  mass  m  and  normal 
thickness  dn(  =  BC).  It  is  acted  upon  by  the  pressures  on 
AD  and  BC,  a  pressure  of  intensity  p  on  the  area  AB(=  a), 
a  pressure  of  intensity  /  -f-  dp  on  the  area  CD,  its  weight 


i8  ROTATION  OF  FLUIDS. 

inclined  at  an  angle  a  to  the  normal,  and  the  centrifugal  force 

m— ,  r  being  the  radius  of  curvature. 

. '  i  >•  • 

-0,. 


Resolving  along  the  normal, 


or 


or 


a  .  dp  —  ;;/  —  —  mg  cos  a  =  o, 


/z>2  \        iva  .  dn  tv2  \ 

a.  dp--  m{—  +  g  cos  aj  =  -  —  —  {—  +  g  cos  orj, 


dp        wlv*  \ 

--7-  =  —  I r-  g  cos  orJ. 

dn       g  \  r  I 

If  the  stream-line  is  in   a  horizontal  plane,  a  —  90°,  and 
then, 

dp    _  w  v2 
dn        cf    r 

But  by  equation  (4),  Art.  4,  since  z  is  now  constant, 


dH      ^_   dp      v   dv_Wv      dv\_2v   ffv      dv\ 

dn  ~~  w  "  dn  '   g "  dn  ~  g\r  ~*  dn^  " "  g  *  2  T  """  dn/  * 


WHIRLING  FLUIDS.  l& 

i  fv       dv\ 
/The    expression    —I- — h  T~J    *s    designated     the    average 

angular  velocity,  or  the  rotation  of  the  fluid. 

Again,  if  the  stream-line  is  horizontal  and  is  also  circular,., 
dn  —  dr,  and 

dp       wv2 

dr  ~~  g  "r " ' 

a  differential  equation  connecting  the  pressure  and  the  velocity.. 
If  v  is  a  known  function  of  r,  the  pressure  can  be  at  once 
determined. 

7.  Whirling  Fluids. — Let  a  fluid  mass  whirl  like  a  rigid 
body  about  a  vertical  axis  YY,  with  an  angular  velocity  GO. 

Consider  the  relative  equilibrium  of  an  element  of  mass  im 


v  ^  v 


FIG.  6. 

at  P  distant  x  horizontally  from  the  axis  and  y  vertically  from- 
the  origin  O  in  YY. 

Take  PA  horizontally  to  represent  the  centrifugal  force 
maPx,  and  PB  vertically  to  represent  the  weight  mg.  The 
remaining  forces  must  be  equal  and  opposite  to  the  resultant  of 
these  two  forces,  viz.,  the  diagonal  PC  of  the  parallelogram 
AB.  The  magnitude  of  this  resultant  is 


PC  = 


20 


LEVEL  SURFACES. 


Its  slope,  a,  is  given  by 
Integrating, 


dy  mg 

-j-  •=.  tan  a  = 5 

dx 


=._£_ 

moj^x     '    GO*X' 


c  being  a  constant  of  integration. 

Thus  an  infinite  number  of  logarithmic  curves  can  be  drawn 
such  that  the  tangent  at  any  point  in  any  one  of  the  curves  is 
in  the  direction  of  the  resultant  force  at  that  point.  These 
curves  are  called  lines  of  force,  and  the  surfaces  cutting  these 
lines  of  force  orthogonally  are  designated  level  or  equipotential 
surfaces. 

If  ft  is  the  slope  of  a  level  surface,  then 


dy 

4-  -~  —  tan  6  —  cot  a  = = 

dx  *"  ^ 


mg 


Integrating, 


c  being  a  constant  of  integration. 

Thus  the  level  surfaces  are  paraboloids  of  revolution. 

For  the  free  surface  this  result  is  obtained  more  simply  as 
follows :   The  fluid  element  of  mass  m  in  the  free  surface  at  P 


is  kept  in  relative  equilibrium  by  (a)  the  centrifugal  force  m&Px 
(£)its  weight  mg,  and  (c]  the  fluid  pressures,  which  must  neces 


EXAMPLES.  21 

sarily  have  a  resultant  normal  to  the  free  surface  at  P.  Draw- 
ing-j:he  horizontal  PN  and  the  normal  PG  to  meet  the  axis  of 
rotation  in  N  and  G,  PNG  is  evidently  a  triangle  of  forces,. 

NG          mg         NG  g 

and  therefore  -7T1r7  = =-  = ,  and  NG  =  — :.,  a  constant. 

PN       mz&x         x  GO* 


Thus,  the  sub-normal  is  constant,  and  the  free  surface  must  be 
a  paraboloid  with  its  vertex  at  the  point  O  where  the  free 
surface  cuts  the  axis  of  rotation. 

Ex.  i.  Deduce  the  law  of  pressure  variation  (a)  for  water  in  a  vessel 
moving  slowly  towards  a  hole  in  the  centre,  the  stream-lines  being  ap- 
proximately horizontal  circles  and  the  velocity  of  any  fluid  particle 
inversely  as  its  distance  from  the  axis  (b)  for  water  rotating  as  a  rigid 
body  about  an  axis  (as  in  a  full  centrifugal  pump  before  delivery  com- 
mences), the  velocity  of  any  fluid  particle  being  directly  proportional  to- 
its  distance  from  the  axis. 

(a)  Take  v  =  — ,  then 

i  dp         i  if_  _  £  a^ 
vj  dr  ~  g    r  ~  g  r3 ' 

Therefore  —  =  c ^-  =  ^  -  — . 

TV  2F  r  2Pr 

(b)  Take   v  =  br,  then 

*x  -  "•'} ' 
\  dp       i   v*        i  „ 

-  -r-  = =  —b*r. 

iv  dr       g  r        g 

6  i  z/a 

Therefore  £.  =  *>+  — £»ra  =<:'+—. 

W  2g  2g 

Ex.  2.  A  cylindrical  vessel,  10  ft.  in  height  and  i  ft.  in  diam.,  is  half 
full  of  water.  Find  the  number  of  revolutions  per  minute  which  the 
vessel  must  make  so  that  the  water  may  just  reach  the  top,  the  axis  of 
revolution  being  coincident  with  (a)  the  axis  of  the  vessel,  (b)  a  gen- 
erating line. 

(a)  The  free  surface  of  the  water  is  the  paraboloid  POP,  Fig.  8,  with 
its  vertex  at  O,  since  the  vol.  of  the  paraboloid 

=  £  vol.  of  circumscribing  cylinder, 
=  vol.  of  water  in  vessel. 

g  latus  rectum        I    PN*        I 

Then  &~  —  NG  = = ^— =•  =  5-, 

cos  22     ON       80 

and  GO  =  4/32  x  80  =  161/10. 


SHARP-EDGED  ORIFICES. 


The  linear  speed  of  the  rim  at  P  =  J&?  =  8|/io, 

.and  the  number  of  revols.  per  min.    =  — - — H—  =  482.96. 

-y-  x  i 


N 


FIG.  8. 


FIG.  9. 


(6)  The  free  surface  is  now  the  paraboloid  OP,  with  its  vertex  at  6>, 
Fig.  9. 

g         i  /Wa         i 
^   =  2"  C?Ar  ~  ' 


Then 


.and 


a?  =  4/640  = 


Thus  the  number  of  revols.  per  min.  = 


20 


8.  Orifice  in  a  Thin  Plate.  —  If  an  opening  is  made  in  the 
"wall  or  bottom  of  a  tank  containing  water,  the  fluid  particles 
immediately  move  towards  the  opening,  and  arrive  there  with 
a  velocity  depending  upon  its  depth  below  the  free  surface. 
The  opening  is  termed  an  "  orifice  in  a  thin  plate,"  when  the 
"water-  springs  clear  from  the  inner  edge,  and  escapes  without 
.again  touching  the  sides  of  the  orifice.  This  occurs  when  the 
bounding  surface  is  changed  to  a  sharp  edge,  as  in  Fig.  10, 
.and  also  when  the  ratio  of  the  thickness  of  the  bounding  sur- 
face to  the  least  transverse  dimension  of  the  orifice,  does  not 


SHARP-EDGED   ORIFICES. 


<exceed  a  certain  amount  which  is  usually  fixed  at  unity,  as  in 
Figs.   1 1  and  12. 

Owing  to  the  inertia  acquired  by  the  fluid  filaments,  there 
will  be  no  sudden  change  in  their  direction  at  the  edge  of  the 
orifice,  and  they  will  continue  to  converge  to  a  point  a  little 
in  front  of  the  orifice,  where  the  jet  is  observed  to  contract  to 
the  smallest  section.  This  portion  of  the  jet  is  called  the  vena 


^contracta,  or  contracted  vein,  and  the  fluid  filamentsfl  ow  across 
the  minimum  section  in  sensibly  parallel  lines,  so  that  here,  if 
the  motion  is  steady,  Bernoulli's  theorem  is  applicable. 

The  dimensions  of  the  contracted  section  and  its  distance 
from  the  orifice  depend  upon  the  form  and  dimensions  of  the 
orifice  and  upon  the  head  of  water  over  the  orifice. 

Let  Fig.  1 3  represent  the  portion  of  the  jet  between  a  cir- 
cular   orifice   of  diameter  AB  and  the    contracted 
•    section    of  diameter    CD,   EF  being   the    distance 
between  AB  and  CD.      Then,   taking  the  average 
P  results  of  a  number  of  observations,  it  is  found  that 

AB,  CD  and  EF  are  in  the  ratios  of  100  to  80  to 
> 
50. 

FIG.  13.  Thus  the  areas  of  the  contracted  section  and  of 

the  orifice  are  in  the  ratio  of  16  to  25,  and,  generally  speaking, 
this  is  assumed  to  be  the  ratio  whatever  may  be  the  form  of 
the  orifice. 


TORRICELLI'S   THEOREM. 


9.  Torricelli's  Theorem. — Let  Fig.  14  represent  a  jet 
issuing  from  a  thin-plate  orifice  in  the  side  of  a  vessel  contain- 
ing water  kept  at  a  constant  level  AB 

Let  XX  be  the  datum  line,  MN  the  contracted  section,  and 
consider  any  stream-line  mny  m  being  in  a  region  where  the 


FIG.   14. 

velocity   is   sensibly   zero,    and   n   in   the    contracted   section. 
Then,  by  Bernouilli's  theorem,  the  motion  being  steady, 


A 


•P 


?i  +  —  +  —  =  2  +  —  -f  — ' 

W          2g  ""'          °  °" 


(0 


/,  /x  being  the  pressures  at  w  and  m,  and  #,  ^.  their  elevations 
above  datum.      Hence 


-    =  z,  —  z  -4- 
2g 

If  the  flow  is  into  the  atmosphere, 

'  p  =  the  atmospheric  pressure 
A  =  w 


p         p 


(2V 


,  and 


O  being  the  point  in  which  the  vertical  through  m  intersects. 
the  free  surface.     Thus 


.     (3) 


li  being"  the  depth  of  n  below  the  free  surface. 


TORRICELLI'S    THEOREM. 


The  result  given  by  equation  (3)  was  first  deduced  by 
Totricelli. 

The  depth  below  the  free  surface  is  very  nearly  the  same 
for  .all  points  of  the  contracted  vein,  and  the  value  of  v  as 
given  by  (3)  is  taken  to  be  the  theoretical  mean  velocity  of 
flow  across  the  contracted  section. 

Equation  (3)  is  equivalent  to  the  statement  that  when  the 
orifice  is  opened,  the  hydrostatic  energy  of  the  water,  viz., 
li  ft.-lbs.  per  pound,  is  converted  into  the  kinetic  energy  of 

V2 
• —  ft.-lbs.   per  pound.      Thus,  if  the  jet  is  directed  vertically 

upwards,  it  will  very  nearly  rise  to  the  level  of  the  free  surface, 
and  would  reach  that  level  were  it  not  for  air  resistance,  or  for 
viscosity,  or  for  friction  against  the  sides  of  the  orifice,  or  for  a 
combination  of  these  retarding  causes. 

If  the  jet  issues  in  any  other  direction,  it  describes  a  para- 
bolic arc  of  which  the  directrix  lies  in  the  free  surface. 

Let  OTV>  Fig.  15,  be  such  a  jet,  its  direction  at  the  orifice 
at  O  making  an  angle  a  with  the  vertical.  With  a  properly 


FIG.  15. 

formed  orifice  a  greater  or  less  length  of  the  jet  will  have  the 
appearance  of  a  glass  rod,  and  if  this  portion  were  suddenly 
solidified  and  supported  at  the  ends,  it  would  stand  as  an  arch 
without  any  shearing  stress  in  normal  sections 


26 


VESSELS  IN  MOTION. 


Again,  the  horizontal  component  of  the  velocity  of  flow  at 
any  point  of  the  jet  is  constant  (=  v  sin  «-),  so  that,  for  the 
unbroken  portion  of  the  jet,  equidistant  vertical  planes  will 
intercept  equal  amounts. of  water,  and  the  height  of  the  C.  G. 
of  the  jet  above  the  horizontal  line  OV,  will  be  two  thirds  of 
the  height  of  the  jet. 

10.  Efflux  through  an  Orifice  in  the  Bottom  or  in  the 
Side  of  a  Vessel  in  Motion. — If  a  vessel  containing  water  z  ft. 
deep  ascend  or  descend  vertically  with  an  acceleration  /,  the 
pressure  /  at  the  bottom  is  given  by  the  equation 


0  being  the  atmospheric  pressure.      Therefore 


q 

Before  an  orifice  is  opened,  if 
the  heavier  vessel  is  reduced  to  rest 
by  applying  an  upward  acceleration 
f,  the  pressure  at  the  depth  z  is 

changed     from    ws    to    wz\l  -\-  —  j, 

O  ' 

while  in  the  other  vessel  it  would  be 
changed  from  ws  to  wzll  —  -j. 

If  now  an  orifice  is  opened  at  the  bottom,  the  velocity  of 
efflux  v  is  still  taken  as  being  due  to  the  head  of  the  pressure 
/,  and  therefore  by  Torncelli's  Theorem 


FIG.  16. 


=  21  ±  - 


f\ 
. 
' 


Let  Wl  be  the  weight  of  the  vessel  and  water,  and  let  the 
vessel    be    connected    with    a   counterpoise   of  weight    W2  by 


FLOW  IN  FRICTION  LESS  PIPE. 


27 


means  of  a  rope  passing  over  a  pulley.      Then  by  Newton's 
second  law  of  motion,  and  neglecting  pulley  friction, 


f 


T 


W, 


T  being  the  tension  of  the  rope.    Therefore,  also,  T=  l 

Next  let  a  cylindrical  vessel, 
Fig.  17,  of  radius  r  and  containing 
water,  rotate  with  an  angular  ve- 
locity oo  about  its  axis,  Art.  7.  The 
surface  of  the  water  assumes  the 
form  of  a  paraboloid  with  its  vertex 
at  O  and  its  latus  rectum  equal 

2g 

to   — -.      If  an  orifice  is  made  at  Q 

CtT 

in  the  side  of  the  vessel,  at  a  verti-  FIG.  17. 

cal  distance  z  from  O,  the  water  will  flow  out  with  a  velocity 

7'  due  to  the  head  of  pressure  at  the  orifice.      This  head  is  PQ, 

and 


±  z. 


the  sign  being  plus  or  minus ,  according  as  the  orifice  is  below 
or  above  O.      Hence,  by  Torricelli's  theorem, 


or 


V2  = 


2gZ. 


ii.  Application  to  the  Flow  through  a  Frictionless  Pipe 
t)f  Gradually  Changing  Section  (Fig.  18). — Let  the  pipe  be 
supplied  from  a  mass  of  water  of  which  the  free  surface  is  H  ft. 
above  datum. 


28  FLOW  IN  FRICTION  LESS  PIPE. 

Let  0j ,  /! ,  ^  be  the  sectional  area,  pressure,  and  velocity- 
of  flow  at  any  point  A ,  ^  ft.  above  datum 
and  /*j  ft.  below  the  free  surface. 

Let  #2 ,  /2 ,  v2  be  similar  symbols  for  a  second  point  B> 
z^  ft.  above  datum  and  /i2  ft.  below  the 
free  surface. 


FIG.  18. 
Then  by  the  condition  of  continuity, 


and  by  Torricelli's  theorem, 


A  -A 


and 


*,  2 


Hence 


o  tr 


A 
w 


HYDRAULIC  COEFFICIENTS. 


29 


so  that  Bernouilli's  theorem,  viz., 


2  <r 


—  //"  4-  -°   =  a  constant. 


holds  true  for  the  assumed  conditions. 

12.  Hydraulic  Coefficients^ — These  are  coefficients  intro- 
duced to  correct  the  discrepancies  between  the  results  deduced 
by  theoretical  considerations  and  the  actual  results  of  practice. 

Numerous  experiments  have  been  made  to  determine  the  values  of 
these  coefficients,  and  with  the  same  object  in  view,  special  apparatus 
has  been  designed  and  installed  in  the  hydraulic  laboratory  of  McGill 
University.  A  main  feature  of  this  apparatus  is  a  cast-iron  tank,  square 
in  section,  28  ft.  in  height,  and  having  a  sectional  area  of  25  sq.  ft. 
Care  has  been  taken  to  make  the  inside  surfaces  of  the  tank  perfectly 
flush,  and  to  this  end  the  flanges,  by  which  the  several  sections  are 
bolted  together,  are  placed  on  the  outside. 

The  valve,  Fig.  19,  in  the  side  of  the  tank  is  a  gun-metal  disc  £  in.  in 
thickness  and  24  ins.  in  diameter,  fitted  into  a  recess  of  the  same  di- 


CAP  FOR 
CHANCING  ORIFICES 


BACK  OR  INSIDE  VIEW 


FIG.  19. 

mensions  in  a  cast-iron  cover-plate,  with  gun-metal  bearing  faces,  form- 
ing a  water-tight  joint  for  the  face  of  the  disc.  This  cover-plate  is 
bolted  to  an  opening  in  the  front  of  the  tank,  and  the  inner  faces  of 
the  cover-plate  and  disc  are  flush  with  the  inner  surface  of  the  tank. 


30  COEFFICIENT  OF  VELOCITY. 

In  the  disc,  and  on  opposite  sides  of  the  centre,  there  are  two  screwed 
openings,  the  one  3  ins.  and  the  other  7  ins.  in  diameter.  By  rotating 
the  disc  each  opening  can  be  made  concentric  with  a  screwed  7|-in. 
opening  in  the  body  of  the  valve.  The  disc  is  rotated  by  means  of  a 
spindle  through  its  centre,  passing  through  a  gland  in  the  front  of  the 
valve  body,  and  operated  by  a  lever  on  the  outside.  Gun-metal  bushes,, 
with  the  required  orifices,  are  screwed  into  the  disc  openings,  and  when, 
screwed  home  have  their  inner  surfaces  flush  with  the  valve  surface,  and 
therefore  with  the  inside  surface  of  the  tank.  By  means  of  a  simple: 
device,  these  bushes  can  be  readily  removed  and  replaced  by  others, 
without  the  loss  of  more  than  a  pint  of  water.  A  cap  with  a  central 
gland  is  screwed  into  the  7|-in.  opening  of  the  valve  body  and  forms  a 
practically  water-tight  cover.  The  valve  is  rotated  so  as  to  bring  the: 
bush  opposite  the  opening,  and  it  is  then  unscrewed  by  means  of  a 
special  key  projecting  through  the  cap-gland.  The  valve  is  now  turned 
back  until  the  opening  is  closed,  when  the  cap  is  unscrewed,  the  bush 
taken  out,  and  another  put  in  its  place.  The  cap  is  again  screwed  into, 
position,  the  valve  rotated  until  the  openings  in  the  disc  and  tank-side- 
are  concentric,  when  the  bush  is  screwed  home  by  the  key. 

A  gun-metal  bush  screwed  into  each  of  the  two  openings  in  the- 
main  disc,  carries  a  series  of  orifice  plates.  The  larger  bush  takes 
plates  with  openings  up  to  4  ins.  in  diameter,  and  the  smaller  bush  takes 
plates  with  openings  up  to  if  ins.  in  diameter.  The  plates  are  provided 
with  a  shouldered  edge,  which  fits  against  the  corresponding  rim  of  the 
bush,  and  are  screwed  with  the  orifice  in  any  required  position  by  means 
of  an  annular  screwed  ring  fitting  the  interior  surface  of  the  bushing. 
The  orifice  plates  are  gun-metal  discs,  4$  ins.  in  diameter  by  ^  in.  thick 
for  the  large  bush,  and  2  ins.  in  diameter  by  •£  in.  thick  for  the  smalt 
bush. 

The  utmost  care  has  been  taken  to  form  the  orifices  with  the  greatest 
possible  accuracy.  The  orifices  are  worked  in  the  discs  approximately 
to  the  sizes  required,  and  are  then  stamped  out  with  hardened-steel 
punches,  the  sizes  of  which  have  been  determined  with  great  exactness, 
by  means  of  Brown  &  Sharpe  micrometers.  The  diameters  of  the  ori- 
fices are  also  checked  by  a  Rogers'  comparator  and  a  standard  scale. 
In  some  cases  a  discrepancy  has  been  found  between  the  sizes  of  the 
die  and  its  orifice,  but  the  size  obtained  for  the  orifice  is  the  one  which 
has  been  invariably  used  in  the  calculations. 

(a)  Coefficients  of  Velocity. — The  actual  velocity  i>  at  the 
vena  contracta  is  a  little  less  than  \/2gh,  the  theoretical 
velocity  (Art.  9),  and  the  ratio  of  v  to  ^2gh  is  called  the 
coefficient  of  velocity.  Denoting  this  coefficient  by  cv,  then, 


COEFFICIENT  OF  VELOCITY. 


and  the  equations  for  the  velocities  of  discharge  in  the  case  of 
moving  vessels  (Art.  10)  become 


and 


it  =  v.  2(g  ±f)/i     ' 

V*   =   Cv\uW    ±    2gs). 


A  mean  value  of  cv  for- well  formed  simple  orifices  is  .974. 
Assuming  that  the  face  of  the  orifice  is  vertical  and  that  the 
jet  issues  horizontally  with  a  velocity  of  v  ft.  per  second,  under 
a  head  of  //  ft.  of  water,  and  assuming  also  that  in  t  sees.,  a 


FIG.  20. 


fluid  particle  reaches  a  point  y  ft.  measured  vertically  and  x  ft. 
measured  horizontally  from  the  point  of  discharge,  then,  dis- 
regarding the  effect  of  air  resistance  and  other  disturbing 
causes, 


x  = 


y  = 


and  therefore 


X*  2V*  2 

-  =    -    -    =   —C*2gh 

y       g      gv  * 


or 


s-       — 
P..     — - 


32  COEFFICIENT  Or    VELOCITY. 

If  xv ,  jj/1  are  the  coordinates  of  the  fluid  particle  in  any  other 
position,  then,  also, 


Hence 

x 2  -  xa 


i  --  y)' 

which  is  the  formula  used  in  the  McGill  laboratory  in  the 
experimental  determination  of  coefficients  of  velocity.  The 
position  of  the  jet  is  defined  by  vertical  measurements  from  a 
straight-edge,  supported  horizontally  above  the  jet,  by  a  bracket 
on  the  tank  face  at  one  end,  and  at  the  other  on  a  bearing, 
which  admits  of  a  vertical  adjustment,  Fig.  2 1 . 


FIG.  21. 

The  straight-edge  is  of  machinery-steel,  is  5^  ft.  in  length, 
2j  ins.  in  depth,  f  in.  in  width,  and  is  graduated  so  as  to  give 
the  horizontal  distances  from  the  inner  face  of  the  orifice' plate. 
The  vertical  ordinates  are  measured  by  a  Vernier  caliper 
specially  adapted  for  the  purpose.  The  flat  face  of  the  movable 
limb  is  made  to  rest  upon  the  upper  surface  of  the  straight- 
edge, and  the  caliper-arm  hangs  vertically.  A  bent  piece  of 
\vire,  with  a  needle-point,  is  clamped  to  the  other  limb,  and, 


COEFFICIENT  OF  VELOCITY. 


33 


by  means  of  the  screw  adjustment,  can  be  readily  moved  until 
it  jitst  touches  the  upper  or  lower  surface  of  the  jet. 

By  means  of  the  above  method,  an  extended  series  of 
'experiments  with  i-in.,  ^-in.,  and  i-in.  sharp-edge  orifices,  and 
sunder  heads  varying  from  6  to  20  ft.,  gave  .99  as  the  average 
^value  of  the  coefficient  of  velocity  (cv). 

Let  the  direction  of  the  jet,  Fig.  22,  at  the  point  of  dis- 
charge make  an  angle  a  with  the  horizontal,  and  let^/j^, 


FIG.  22. 


-*-2i  y*i    be   the   coordinates   defining   the    position   of  a  fluid 
particle  after  intervals  of  tl  sees,  and  t2  sees.      Then 


jtrl  =  i'  cos  a  ,  /t        and        jj',  =  i'  sin  a  .  /t  — 
^  =  f  cos  «  .  /.j       and       y^  =  v  sin  a  .  /2  — 
These  equations  give 


tan  »  — 


*!**(**    ~    ^l 


and 


*  sec2  a  2v*  2 

' 


sec2  a 


l  tan  «  -  Jt          g  x.2  tan  a  —  yj 

from  which  a:  and  then  cn  can  be  calculated. 


34          COEFFICIENTS  OF  RESISTANCE  AND  CONTRACTION. 

(b)  Coefficient  of  Resistance.  —  Let  hv  be  the  head  required 
to  produce  the  velocity  v.  Let  hr  be  the  head  required  to> 
overcome  the  frictional  resistance.  Then 

//,  the  total  head,  =  hv  +  hr  —  hv(i  +  <rr), 

where  hr  =  crhv. 

cr  is  termed  the  coefficient  of  resistance,  and  is  approxi- 
mately constant  for  varying  heads  with  simple  sharp-edged 
orifices.  Again, 


Hence 

h  =  c*h(\  +  cr)t 
and  therefore 

7?     =      I     +    Cr', 

cv 

so  that  cr  can  be  found  when  cv  is  known,  and  vice  versa. 

(c)  Coefficient  of  Contraction.  —  The  ratio  of  the  area  a  oF 
the  vena  contracta  to  the  area  A   of  the  orifice  is  called  the 
coefficient  of  contraction,  and  may  be  denoted  by  cc. 

The  value  of  cc  must  be  determined  in  each  case,  but  in 
sharp-edged  orifices  an  average  value  of  cc  ,  as  already  pointed 

16 
out,  is  —  =  .64.      Cceteris  paribus,  cc  increases  as  the  orifice 

area  and  the  head  diminish. 

The  following  are  some  of  the  conditions  which  tend  to- 
modify  the  value  of  cc  : 

(i)  The  contraction  is  imperfect  and  will  be  suppressed 
over  the  lower  edge  of  a  square  orifice  at  the  bottom  of  a 
vessel,  and  over  a  side  as  well  if  the  orifice  is  in  a  corner.  In 
fact,  the  contraction  is  more  or  less  imperfect  for  arty  orifice 


COEFFICIENT  OF  CONTRACTION. 


35 


FIG.  23. 


within  three   diameters  from  the  side  or  bottom  of  the  vessel. 
Thus,  the  cross-section  of  the  vena  contracta  is 
increased,   and  experiment   shows  that  the  dis- 
charge is  also  increased. 

(2)  cc  is  increased  or  diminished  according  as 
the  surface  surrounding  the  orifice  is  convex  or 
concave  to  the  interior  of  the  vessel. 

(3)  The  contraction  is  imperfect  and  cc  is  increased,  if  the 
orifice  is  placed  in  a  confined  part  of  the  vessel,  or  if  the  water 
approaches  the  orifice  through  a  channel,  as  in  Fig.  23,  the 
velocity    of  the    fluid     filaments    being    thereby    considerably 
increased. 

(4)  If  the  inner  edge  of  an  orifice  is  rounded,  as  shown  by- 
Figs.    24  and   25,  the   contraction   is   more   or  less  imperfect. 


FIG.  24. 


FIG.  25. 


The  value  of  cc  varies   from   .64  for  a  sharp-edged  orifice  to 
very  nearly  unity  for  a  perfectly  rounded  orifice. 

(5)  The  contraction  is  incomplete  when  a  border  or  rim  is 
placed  round  a  part  of  the  edge  of  the  orifice,  projecting 
inwards  or  outwards.  According  to  Bidone, 

ct  =  .62(1  -f-  .I52-J  for  rectangular  orifices, 


36 
and 


COEFFICIENT  OF  CONTRACTION. 


-[-  .  128— j  for  circular  orifices, 


n  being  the   length   of  the   edge  of  the  orifice  over  which  the 
border  extends,  and  p  the  perimeter  of  the  orifice. 

(6)   If  the  sides   of  the   orifice  are  curved  so  as  to  form   a 
bell-mouth   of  the  proportions  shown  by  Fig.  26,  and   corre- 


FIG.  26. 

spending  approximately  to  the  shape  of  the  vena  contracta, 
the  whole  of  the  contraction  will  take  place  within  the  bell- 
mouth,  and  cc  is  unity  if  the  area  of  the  orifice  is  taken  to  be 
the  area  of  the  smaller  end. 

For  such  an  orifice  Weisbach  gives  the  following  table  of 
values  of  cv : 

Head  over  Orifice  in  Feet.  cv. 

•66 .   .959 

1.64 967 

11-48.  .  975 

55-77-  994 

337-93 994 

The  dimensions  of  the  jet  at  the  con- 
tracted section  or  at  any  other  point,  may 
be  directly  measured  by  means  of  set- 
screws  of  fine  pitch,  arranged  as  in  Fig. 
27.  The  screws  are  adjusted  so  as  to 
touch  the  surface  of  the  jet,  and  the  dis- 
tance between  the  screw-points  is  then 
FIG.  17.  measured. 


COEFFICIENT  OF  CONTRACTION. 


37 


Measurements  of  very  great  accuracy  can  be  made  with  the  jet- 
measurer,  Fig.  28,  designed  and  constructed  in  the  McGill  laboratories 
which  may  be  described  as  follows — One  end  of  a  horizontal  2-in.  bar 


FIG.  28. 

is  attached  to  the  front  of  the  tank  (Fig.  28)  and  the  other  is  supported 
on  a  frame  bolted  to  the  sides  of  the  flume.  A  split  sleeve  slides  along 
this  bar  and  may  be  clamped  by  a  tightening  screw  in  any  desired  posi- 
tion. Upon  the  cross-head  there  is  another  split  boss,  or  sleeve,  through 
which  a  second  bar  passes  at  right  angles  to  the  first,  and  carries  a  sim- 
ilar cross-head  to  that  on  the  2-in.  bar,  so  that  provision  is  made  for  a 
rough  adjustment  in  a  vertical  plane.  Through  the  latter  cross-head 
passes  a  smaller  bar.  and  along  this  bar  slides  a  third  adjustable  cross- 
head,  or  caliper-holder,  by  which  the  caliper  can  be  swung  round  and 
receive  its  final  adjustment.  For  the  measurements  a  i2-in.  Brown  & 
Sharpe  vernier  caliper  is  used.  A  capstan  head  rod  is  clamped  to  each 
leg  and  can  be  swivelled  through  any  angle.  Steel  needle-pointers  are 
inserted  in  the  heads,  and  are  clamped  in  such  position  as  may  be  re- 
quired. In  making  a  measurement  the  steel  points  are  first  made  ta 


COEFFICIENT  OF  DISCHARGE. 


touch  and  the  corresponding  readings  taken.  The  points  are  then  sep- 
arated by  sliding  the  caliper-heads  apart,  and  the  whole  apparatus  is 
moved  into  position.  The  points  are  finally  adjusted  so  as  to  touch  the 
surfaces  of  the  jet  at  opposite  points,  and  readings  are  again  taken. 
From  the  two  sets  of  readings  the  transverse  dimension  of  the  jet  can  be 
at  once  determined,  to  the  one-thousandth  of  an  inch,  and  at  any  point 
between  72  ins.  and  ^  in.  from  the  inner  surface  of  the  orifice-plate. 
Rigidity  in  the  apparatus  is,  however,  most  essential. 

(d}    Coefficient    of  Discharge. — If   Q  is   the   discharge    in 
cubic  feet  per  second  across  the  contracted  section,  then 
Q  =  av  ==^ty/^/i  --=  cA  V^h, 

where  c  —  cccv,  is  the  coefficient  of  discharge  and  is  to  be 
determined  by  experiment. 


FIG.  29. 

In  the  experiments  made  in  the  McGill  laboratory,  the  water,  on  leav- 
ing the  orifice,  passes  either  to  waste,  or  to  the  measuring  tank  through 
a  bifurcated  galvanized-iron  tubing,  supported  in  a  pivoted  frame, 
Fig.  29.  The  water  is  first  run  to  waste  through  one  of  the  branches 
until  a  steady  head  has  been  obtained,  and  the  frame  is  then  rapidly- 
swung  through  a  small  angle  by  means  of  a  lever,  when  the  water  passes 
through  the  other  branch  to  the  tank.  As  soon  as  the  tank  is.  sum*- 


COEFFICIENT  OF  DISCHARGE.   . 


39 


•ciently  full,  the  frame  is  swung  back  and  the  water  again  runs  to  waste. 
At  first  the  water  discharged  from  the  tank  was  replaced  by  water  ad- 
mitted into  the  top  of  the  tank  through  a  hose  terminating  in  a  rose 
submerged  just  below  the  surface.  Aitnougii  the  utmost  care  had  been 
taken  in  the  design  of  this  rose  to  reduce  the  eddy  motion  at  efflux  to  a 
minimum,  there  was  always  an  appreciable  disturbance.  The  hose  was 
therefore  extended  until  the  rose  lested  on  tne  bottom  of  the  tank,  8 
feet  below  the  orifice  ,  with  this  arrangement  a  series  of  orifice-flow 
experiments  were  made,  the  time  in  each  case  being  the  mean  01  that 
given  by  two  stop-watches  and  the  values  of  the  coefficients  of  dis- 
charge are  given  in  Tables  A  and  B. 

TABLE  A. 

TRIANGULAR    ORIFICE    OF  .05    SQ.  IN.  AREA  AND   REMAINING 
ORIFICES    OF    .0625   SQ.   IN.  AREA. 


Head 
in 

Feet. 

Circular. 

Equilateral  Tri- 
angle with 
Horizontal  Base 

Square  with 
Vertical  Sides. 

Rectangle  with 
Vertical  Sides 
Equal  to  Four 

Rectangle  with 
Vertical  Sides 
Equal  to  Sixteen 

Uppermost. 

Timesthe  Width 

Times  the  Width 

T 

S 

T 

S 

T 

S 

T 

S 

T 

S 

I 

.678 

.620 

•657 

.63: 

-643 

.627 

.662 

.640 

.688 

.67I 

2 

.6l3 

.613 

.646 

.623 

.63. 

.621 

.643 

.629 

•655 

.657 

4 

.6lO 

.605 

.628 

.616 

.620 

.615 

.631 

.620' 

.642 

.643 

6 

.607 

.601 

.628 

•613 

.615 

.612 

.627 

.616 

•634 

.636 

8 

.606 

.601 

.621 

.610 

•613 

.609 

.624 

.613 

.631 

•  632 

10 

.604 

.600 

.618 

.608 

.612 

.608 

.621 

.613 

.629 

.629 

12 

•603 

098 

.617 

.607 

,6ll 

.607 

.621 

•  6ll 

.626 

.627 

14 

.602 

.598 

.6l7 

.607 

•  6lO 

.606 

.620 

.6lO 

.623 

.625 

16 

.602 

.598 

.616 

.606 

.609 

.606 

.619 

.609 

.622 

.625 

18 

.601 

•597 

.615 

.605 

.607 

.605 

.618 

.608 

.622 

.623 

20 

.601 

•597 

6l5 

•  605 

.607 

.604 

.618 

.608 

.621 

,622 

The  presence  of  the  hose  in  the  tank  was  not  satisfactory,  as  it  neces- 
sarily interfered  with  tne  stream-line  motion,  and  therefore  affected  to  a 
greater  or  less  extent  the  values  of  the  coefficient  of  discharge.  The 
hose  was  discarded,  and  the  water  is  now  admitted  into  a  3-inch  cham- 
ber extending  right  across  the  bottom  of  the  tank  and  containing  per- 
forations on  the  lower  surface  through  which  the  water  flows  to  the 
bottom  and  is  there  deflected  upwards.  Twelve  inches  above  the  bottom 
the  water  is  made  to  pass  through  a  baffle-plate  perforated  with  f-in. 
holes,  and  6  inches  higher  there  is  a  second  baffle-pJate  also  perforated 
with  f-in.  holes.  In  order  to  equalize  as  much  as  possible  the  flow  from 
ail  points,  the  pitch  of  the  holes  in  the  upper  plate  was  determined  by 
the  projections  on  a  horizontal  plane  of  equal  distances  on  a  sphere  of 
10  ft.  diar.  with  its  centre  at  the  centre  of  the  orifice  of  discharge. 

There  are  two  outlet  pipes  lor  fast  and  slow  discharge,  and  there  are 
two  inflow  pipes,  the  one  3  ins.  and  the  other  i|  ins.  in  diameter.  Each 
•of  these  pipes  is  controlled  by  a  stop-valve. 


COEFFICIENT  OF  DISCHARGE. 


TABLE  B. 

ORIFICES    OF    .197    SQ.    IN.    AREA, 


Rectang. 

Rectang. 

Rectang.. 

Eoui  lateral 

Rectangle 

with 

with 

with 

Head 
in 
Feet. 

Circular. 

Triangle 
with  Hori- 
zontal Side 
Uppermost. 

Square 
with 
Vertical 
Sides. 

Square 
with 
Diagonal 
Vertical. 

with  Verti- 
cal Sides 
Equal  to 
Four  Times 
the  Width. 

Vertical 
Sides 
Equal  to 
One- 
quarter 

Vertical 
Sides 
Equal  to 
Sixteen 
Times 

Vertical 
Sides 
Equal   to 
One  six- 
teenth 

Width. 

Width. 

Width. 

T         S 

T         S 

T         S 

S 

T           S 

S 

S 

S 

I 

.624    .618 

.627    .627 

.623    .628 

.623 

.635    .640 

.641 

.658 

.659, 

2 

.616    .6ll 

.620    .621 

.613    .621 

.619 

.626    .633 

•632 

.646 

.646- 

4 

.610    ,607 

.615    .615 

.606    .617 

.614 

.619    .629 

.629 

.637 

•637 

6 

.  607    .  605 

.613    .613 

.604    .614 

.612 

.616    .625 

.627 

•634 

.633 

8 

.  606    .  604 

.612    .6l2 

.603    .612 

.612 

.614    .625 

.625 

.631 

.631 

TO 

.  606    .  604 

.6ll    .6ll 

.602    .6lO 

.611 

.612    .624 

.623 

.630 

.629 

12 

.  605    .  603 

.611    .6ll 

.601    .610 

.611 

.611    .622 

.622 

.627 

.626. 

14 

.604    .603 

.6iO    .6lO 

.600    .610 

.609 

.6ll    .622 

.621 

.624 

.625 

16 

.  606    -  6O2 

.610    .610 

.600   .6lO 

.609 

.6lO    .620 

.621 

.624 

.624 

18 

.  605    •  602 

•6lO    .6lO 

.600    .610 

.609 

.609    .620 

.620 

.623 

.623 

2O 

.604    .601 

.  609    .  609 

.600    .610 

.609 

.  602    .  620 

.620 

.622 

.622. 

S         N.B. — In  Tables  A  and  B,  T  indicates  a  thickness  of  plate  of  .i6-in.,  ) 
(  and  S  indicates  that  the  orifice  is  sharp-edged.  C 

The  time  is  also  measured  electrically.     In  the  forward  and  returns 
movements,  the  lever,  controlling  the  angular  movement  of  the  galvan- 
ized-iron  tubing,  makes  and  breaks  an  electric  contact,  so  that  the  inter 
val  of  time  occupied  by  an  experiment  is  registered  on  a  chronograph.. 

With  this  new  arrangement,  the  following  values  for  the  coefficient 
of  discharge  have  been  deduced  for  sharp-edged  orifices,  the  area  in  each 
case  being  practically  the  same,  viz.,  .19635  sq.  ins.,  and  equivalent  to 
the  area  of  a  circle  of  in.  diameter  : — 


Rectangular 

Rectangular 

Head 

qua, 

Ratio  of  Sides  4  :  i. 

Ratio  of  Sides  16  :  i. 

Trian- 

in 

Circular. 

gular 

Feet. 

Sides 

Diagonals 

Long  Side 

Long  Side 

Long  Side 

Long  Side 

Vertical. 

Vertical. 

Vertical. 

Horizontal 

Vertica". 

Horizontal 

I 

.6199 

.6267 

.6276 

.6419 

.6430 

.6633 

.6644 

.6359- 

2 

.6131 

.6204 

.6277 

•6335 

.6355 

•6503 

.6510 

.6280 

4 

.6081 

.6162 

.6177 

.6281 

.6293 

.6409 

.6415 

.6228 

6 

.6073 

.6137 

.6156 

•6255 

.6266 

.6368 

.6372 

.6202 

8 

.6056 

.6127 

.6138 

.6234 

.6252 

.6342 

.6346 

.6*89 

10 

.6050 

.6ll6 

.6132 

.6224 

.6240 

.6323 

•6327 

.6183 

12 

.6040 

.6109 

.6123 

.6217 

.6230 

.6311 

.6314 

.6177 

14 

.6038 

.6104 

.6ll8 

.6207 

.6222 

.6304 

.6304 

.6176 

16 

.6032 

-.6099 

.6113 

.6203 

.6215 

.6301 

.6298 

.6171 

18 

.6031 

.6096 

.6110 

.6200 

.6212 

.6299 

.6293 

.6163 

20 

.6029 

.6094 

.6108 

.6198 

.62IO 

.6291 

.6285 

.6160 

COEFFICIENT   OF  DISCHARGE.  41 

At  least  two  sets  of  measurements  were  made  for  each 
head,  and  the  mean  was  adopted  as  correct,  if  the  results  d'd 
not  differ  by  more  than  3  in  10,000. 

Numerous  experiments  with  a  i-in.  sharp-edged  orifice 
give  .6  as  an  average  value  of  the  coefficient  of  discharge  for 
heads  varying  from  I  to  20  ft. 

The  jet  springs  clear  from  the  orifice  in  all  cases  repre- 
sented in  the  above  tables,  and  the  following  inferences  may 
be  drawn  from  an  inspection  of  the  same:— 

(1)  The    coefficient   of  discharge   diminishes   as   the   head 
increases,  but  at  a  diminishing  rate. 

(2)  The  coefficients  for  the  thick-plate  orifices  are  in  all 
cases    greater  than   the   corresponding  coefficients   for   sharp- 
edged  orifices,  excepting  in  the  case  of  the  longest  rectangular 
orifice.      Under  a  head  of  I  ft.  the  coefficient  of  discharge  for 
this  orifice  still  exceeds  that  of  the  same  orifice  with  a  sharp 
edge,  while  for  heads  exceeding  I  ft.  the  coefficient  seems  to  be 
a  little  less,  but  is  practically  the  same.      It  may  be  noted  that 
the  thickness  of  the  plate  is  2.56  times  the  width  of  the  orifice,, 
and  the  contraction  for  the  thick-plate  orifice  is  consequently 
increased. 

(3)  The   coefficient   for   rectangular   orifices    seems    to   be 
practically  the  same  whether  the    longest   side   is  vertical   or 
horizontal. 

(4)  The  coefficient  increases  with  the  area  of  the  orifice,, 
excepting  when  the  head  is  very  small.      The  coefficient  for 
orifices  of  small  area  then  rapidly  increases. 

(5)  With  rectangular  orifices,   the  coefficient  increases  as 
the  width  of  the  orifice  diminishes,  i.e.,  as  the  orifice  becomes 
more  elongated. 

The  two  last  results  are  in  accordance  with  similar  results 
deduced  by  Weisbach,  Buff,  and  others. 

NOTE. — The  manner  in  which  the  head  of  water  in  the  tank  is  defined  is 
both  simple  and  effective.  A  glass  gauge,  of  i^  in.  diar.,  is  fixed  to  the 
tank  by  iron  brackets  and  extends  from  the  top  to  the  bottom.  On  one 


4  ^  EXAMPLES. 

side  of  the  gauge  there  is  a  brass  scale  graduated  from  a  zero  point  in  the 
same  horizontal  plane  as  the  centre  of  the  orifice  of  discharge.  A  car- 
rier, with  a  horizontal  wire  passing  in  front  of  the  gauge,  slides  up  and 
down,  and  any  required  head  is  obtained  by  bringing  the  necessary  scale 
graduation,  the  surface  of  the  water  in  the  gauge,  the  wire  and  its 
reflection  in  a  mirror  at  the  back  of  the  gauge,  into  the  same  hori- 
zontal plane.  There  is  a  second  indicator  on  the  opposite  side  of  the 
tank,  consisting  of  afloat  attached  to  an  ordinary  water-proof  silk  fishing- 
cord  passing  over  a  large  light  frictionless  pulley  and  then  vertically 
-downwards  in  front  of  the  tank.  The  cord  is  kept  taut  by  a  weight  at 
the  bottom,  and  carries  a  friction-tight  pointer  which  can  be  easily  and 
rapidly  adjusted  to  indicate  any  required  mark  on  a  brass  plate  fixed  in 
a  convenient  position  on  the  tank  face,  so  that  the  operator  working  the 
valves  has  it  constantly  under  observation.  As  soon  as  the  head  of 
water  in  the  tank  has  been  determined  by  means  of  the  glass  gauge,  the 
pointer  is  moved  into  position  opposite  the  mark,  and  is  kept  there 
throughout  the  experiment.  This  obviates  the  necessity  of  constantly 
watching  the  level  of  the  water  in  the  gauge,  which,  on  account  of  the 
height  of  the  tank,  is  often  very  inconvenient  and  troublesome.  Occa- 
sionally, however,  it  is  advisable  to  check  the  position  of  the  pointer  by 
observing  the  water-level  in  the  gauge,  as  the  cord  indicator  is  extremely 
sensitive,  and  the  cord  itself  necessarily  varies  slightly  in  length,  so  that 
small  errors  might  otherwise  be  introduced. 

The  head  of  water  is  brought  to  any  required  level  by  means  of  a 
three-way  valve  through  which  the  water  can  either  be  admitted  or 
allowed  to  escape.  The  valve  is  provided  with  a  long  vertical  spin- 
dle, upon  which  handles  are  arranged  at  different  points  in  such  man- 
ner that  one  can  be  easily  reached  and  operated  from  any  position 
in  the  height  of  the  tank.  Close  to  the  cord  indicator  and  within  the 
reach  of  the  operator  there  is  a  small  J-in.  pipe  with  valve,  which  is  useful 
for  a  fine  adjustment  when  the  inflow  is  only  slightly  in  excess  of  the 
discharge. 

Ex.  i.  A  vessel,  6  ft.  in  diar.,  is  full  of  water  and  makes  100  revols. 
per  min.  Find  the  velocity  of  efflux  through  an  orifice  2  ft.  below  the 
surface  of  the  water  at  the  centre,  assuming  the  coefficient  of  velocity  to 
be  unity. 

The  linear  velocity  of  the  vessel's  periphery 

27T  .  IOO          22O  , 

=  3«  =  3  •  — ^—  =  —  ft-  Per  sec-         fl  tO 
Hence  the  velocity  of  efflux 


=/(?)' 


2. 


EXAMPLES.  43 

48,400 


49 


+  128  =  33.4  ft.  per  sec. 


Ex.  2.  The  area  of  an  orifice  in  a  thin  plate  was  36.3  cm.2,  the  dis- 
charge under  a  head  of  3.396  m.  was  found  to  be  .01825  m.8  per  sec.,  and 
the  velocity  of  flow  at  the  contracted  section,  as  determined  by  meas- 
urements of  the  position  of  the  axis  of  the  jet,  was  7.98  m.  per  second. 
Find  the  coefficients  of  velocity,  discharge,  contraction,  and  resistance, 
=  9.81  m. 


Therefore,  7.98  =  cv  4/2  x  9.81   x  3.396, 

and  cv  =  .97729, 

O  =  cA  i/2p^. 


36. 


"Therefore          .01825  =  c  x  f—^  V 2  x  9-8 1  x  3.396, 
and  c  =  -6159, 

- — —  =  .632, 
.97729 


i  /    i    v 

•  ~  *7 '"" '  =  1^9772^ )  "  T  =  '°4 


Ex.  3.  The  jet  from  an  orifice  of  .008  sq.  ft.  area,  under  a  head  of  16 
ft.,  issues  horizontally  and  falls  i  ft.  vertically  in  a  horizontal  range  of 
7.68  ft.  Find  the  coefficient  of  velocity. 

(7-68)2 

=   .9216, 


v 

4  x  i  x  16 
and 


Ex.  4.  If  .625  is  the  coefficient  of  discharge  in  the  preceding  exam- 
ple, find  the  discharge  in  gallons  per  minute.  The  orifice  is  rectangular 
and  is  .2  ft.  wide  by  .04  feet  deep.  Find  the  discharge  when  the  contrac- 
tion is  suppressed  over  the  lower  edge  by  means  of  a  projecting  rim. 


Q,  in  cub.  ft.  per  sec.,  =  .625  x  .008  4/2 .  32  .  16  =  .16, 
and  therefore 

the  discharge  in  gallons  per  minute  =  60  x  .16  x  6± 

=  60. 

When  the  contraction  is  suppressed  over  the  lower  edge, 

/  .2         \        1.9778 

the  coeff.  of  contraction  =  .62   i  +  .152— .     =  

V  3  2(.2  +  .o4)/  3 


44 


Therefore 


MINER'S  INCH. 


the  coeff.  of  discharge  =  .96  x 


1.9778 


=  .632896. 


Hence  the  discharge  in  cubic  feet  per  second 

=  .632896  x  .008  x  |/2  .  32  .  16  =  .162 
=  60,758  gallons  per  minute. 

13.  Miner's  Inch.  (7>.  Can.  Soc.  C.  E.,  1900).— The 
miner's  inch  of  water  is  an  arbitrary  module  adopted  in  mining 
districts  for  selling  water.  It  is  variously  defined  as  being  the 
amount  of  water  discharged  per  minute  by  an  orifice  I  in. 
square,  or  an  equivalent  fraction  of  a  larger  orifice,  with  a 
head  of  from  6  to  9  ins.,  the  thickness  of  the  orifice  being 
usually  2  inches. 


FIG.  30. 

One  great  difficulty  is  that  this  is  a  variable  quantity  de- 
pending upon  the  specified  head,  and  therefore  all  such  mod- 
ules should  also  define  the  flow  in  cubic  feet  per  minute. 

There  are  many  practical  difficulties  in  the  way  of  deliver- 
ing absolutely  exact  quantities  of  water,  but  the  definition  of 
the  module  or  unit  should  be  correct  within  a  reasonable  limit 
of  error.  If  it  is  a  definition  of  a  single  miner's  inch  from  an 
orifice  of  i  sq.  in.,  it  should  go  no  further;  but  if  the  inch  is 
defined  as  being  some  fractional  part  of  the  discharge  from  a. 


MINER'S  INCH.  45 

larger  orifice,  it  should  be  limited  to  the  capacity  of  that  orifice. 
Fturther,  as  it  is  a  term  of  local  signification  only,  the  dis- 
charge should  be  given  in  cubic  feet  per  minute,  convenient 
discharges  being  i|  and  2  cu.  ft.  The  flow  under  low  heads  is 
irregular.  Heads  of  I  ft.  or  more  are  not  suitable,  because 

o 

the  water  is  delivered  from  ditches  or  flumes  in  which  the 
depth  is  never  great. 

The  question  thus  resolves  itself  into  a  choice  of  a  stand- 
ard module  or  unit  from  a  flow  under  one  of  two  conditions, 
viz.  : 

(1)  With   a   low  head   of  6J  ins.  above  the  centre  of  the 
orifice   giving   a  discharge  of  i£  cu.  ft.  per  minute,  with  the 
advantage    that    it    is    already   practically    recognized   as    the 
miner's  inch,  and  with  the  disadvantage  that  the  flow  is  irreg- 
ular. 

(2)  With  a  head  of  1 1 J  ins.  above  the  centre  of  the  orifice, 
and  a  discharge  of  2  cu.  ft.  per  minute,  the  flow  being  much 
more  regular,  but  the  quantity  discharged  is  not  recognized  in 
practice. 

The  flow  under  the  first  condition  is  chosen  as  being  the 
one  now  in  use  in  British  Columbia,  and  the  following  specifica- 
tion is  given  of  the  miner's  inch,  including  discharges  of  from 
I  to  100  miner's  inches  of  i£  cu.  ft.  per  minute: — 

The  water  taken  into  a  ditch  or  sluice  shall  be  measured 
at  the  ditch  or  sluice  head.  It  shall  be  taken  from  the  main 
ditch,  flume,  or  canal,  through  a  box  or  reservoir  arranged  at 
the  side,  and  the  water  shall  have  no  appreciable  velocity  of 
approach.  The  orifice  shall  be  fixed  vertically  at  right  angles 
to  the  delivering  waterway,  and  the  edges  and  corners  shall 
be  square  and  sharp,  the  top,  bottom,  and  sides  of  the  orifice 
being  at  right  angles  to  the  pressure-board.  The  issuing  vein 
shall  be  fully  contracted,  and  the  discharge  shall  pass  freely 
into  the  air.  The  distance  between  the  sides  and  bottom  of  the 
orifice  and  the  sides  and  bottom  of  the  waterway  shall  be  at 
least  three  (3)  times  the  least  dimension  of  the  orifice.  The 
miner's  inch  of  water  shall  mean  Tl¥  of  the  quantity  which  shall 


46 


MINER'S  INCH. 


be  discharged  through  an  orifice  six  (6)  ins.  wide  and  two  (2) 
ins.  high,  made  of  2-in.  planks,  planed,  made  smooth  and 
painted.  The  water  shall  have  a  constant  head  of  6J  ins. 
above  the  centre  of  the  orifice,  and  the  amount  discharged 
shall  be  estimated  at  ij  cu.  ft.  per  minute. 

Discharges  up  to  and  including    101.55   miner's  inches  of 
I J  cu.  ft.  of  water  per  minute  shall  be  as  in  the  following  table  * 


Dimensions  of  Orifice  in  Inches. 

Head  in  Inches 

Numberof  Miner's 

/^-     *         f 

Tnrhf»«s  nf     i    f^iihin 

Width. 

Depth. 

Orifice. 

Feet  per  Minute. 

6 

2 

6.25 

11.9858 

12 

2 

6.25 

24.2485 

18 

2 

6.25 

36-3851 

24 

2 

6.25 

48.6865 

4 

4 

6.25 

15-6998 

6 

4 

6.25 

23.5560 

12 

4 

6.25 

47.2853 

18 

4 

6.25 

71  .6296 

25i 

4 

6.25 

101.5495 

T.  Drummond,  B.A.Sc.,  has  made  an  interesting  series  of 
experiments  (Trans.  Can.  Soc.  C.  E.,  vol.  XIV,  1900)  on  the 
Miner's  Inch,  in  the  Hydraulic  Laboratory,  McGill  University. 

The  discharges  recorded  were  made  under  low  heads   of 
from  6  to   12  ins.,  and  with  two  kinds  of  orifices,  viz. : 

(1)  Standard  sharp-edged  rectangular  orifices  in  brass  from 

1  to  4  sq.  ins.  in  area. 

(2)  Square-edged  rectangular  orifices  in  wood,  2  ins.  thick,, 

2  to  4  ins.  in  height,  and  J-  to  24  ins.  in  width. 

The  formula  adopted  for  the  discharge  was 

Q  =  \CB  tiTg(Hf  -  H*)  (see  Article  22), 
in  which  C  is  the  coefficient  of  discharge ; 
g\*  32.176; 

Q  is  the  discharge  in  cubic  feet  per  second ; 
B  is  the  width  of  the  orifice ; 
HI  and  H2  the  heads  over  the  top  and  bottom  of  the 

orifice. 
No  corrections  were  made  for  changes  in  temperature. 


MINER'S  INCH.  47 

The  shape  of  the  orifice  has  a  very  sensible  effect  upon  the 
discharge.  Circular  orifices  give  the  least  discharge,  the 
greatest  discharges  occur  with  rectangular  orifices,  while  the 
discharges  with  square  orifices  are  intermediate.  The  coefficient 
of  discharge  (C)  diminishes  as  the  size  of  the  orifice  increases, 
the  same  form  of  orifice  being  maintained.  For  the  same 
orifice  C  diminishes  as  the  head  increases.  In  rectangular 
orifices  of  constant  depth  the  coefficient  of  discharge  increases 
with  the  width.  If  the  width  remains  constant,  the  coefficient 
increases  as  the  depth  diminishes. 

These  experiments  illustrate  a  curious  point,  namely,  that 
various  small  orifices,  2  ins.  thick  (made  in  a  2-in.  plank),  run 
full  like  a  short  tube, .and  these  orifices  therefore  discharge 
more  water  than  they  theoretically  should  if  the  vein  were 
contracted.  The  £-in.  X  2-in.,  i-in.  X  2-in.,  and  2-in.  X 
2-in.  orifices  run  full  under  these  conditions,  as  also  does  the 
i-in.  X  i-in.  orifice. 

The  i-in.  X  2-in.  orifice,  2  ins.  thick,  is  just  on  the  margin 
between  flow  with  contraction  and  full-bore  flow.  If  it  is  fixed 
in  the  vertical  position,  with  the  longest  diameter  vertical,  the 
vein  contracts.  If  it  is  fixed  in  the  horizontal  position,  with 
the  longest  diameter  horizontal,  it  will  also  contract,  but  if  it 
is  rubbed  with  the  fingers  on  the  edge,  it  will  run  full  for  a 
time  and  then  contract  again.  If  kept  running  full  in  this  way, 
it  will  discharge  about  I  cu.  ft.  of  water  per  minute  more  than 
when  full  contraction  takes  place. 

The  2-in.  X  2-in.  orifice  runs  partly  full,  that  is  to  say,  the 
lower  half  of  the  orifice,  where  the  issuing  vein  curves  down, 
runs  full,  while  the  upper  half  contracts.  This  largely  in- 
creases both  the  discharge  and  the  coefficient  of  discharge,  but 
the  flow  becomes  irregular  and  it  is  therefore  practically 
impossible  to  measure  a  simple  miner's  inch.  For  this  reason 
y1^  of  the  flow  from  the  6-in.  X  2-in.  orifice  was  chosen  as 
the  standard  for  the  unit  miner's  inch,  and  this  miner's  inch 
actually  discharges  1.4982  cu.  ft.  per  minute. 


48 


INVERSION  OF   THE  JET. 


14.  Inversion  of  the  Jet. — The  phenomenon  of  the  inver- 
sion of  the  jet  was  first  noticed  by  Bidone,  and  has  been 
subsequently  investigated  by  Poncelet,  Lesbros,  Magnus,  Lord 
Rayleigh,  the  author,  and  others. 


Sectional   Elevation. 


Cross-section. 


FIG.  31. 


FIG.  32. 


When  a  jet  issues  from  an  orifice  in  a  vertical  surface,  the 
sections  of  the  jet  at  points  along  its  path  assume  singular 
forms  dependent  upon  the  nature  of  the  orifice. 

With  a  square  or  rectangular  orifice  the  section  of  the  jet 

is  a  star  of  four  sheets  at  right 
angles  to  the  sides,  Figs.  31,  32, 

33- 

With  a  triangular  orifice  the 
section  is  a  star  of  three  sheets 
at  right  angles  to  the  sides, 
Fig.  34- 

In  general,  with  a  polygonal  orifice  of  n  sides,  the  section  is 
a  star  of  n  sheets  at  right  angles  to  the  sides. 


FIG.  33- 


FIG.  34. 


INVERSION  OF   THE  JET.  49 

These  jets  from  non-circular  orifices  have  central  cores,  and 

sheets  at  the  edges  are  thickened  out  into  beads,  Figs.  33, 
and  34,  which  are  approximately  elliptical  in  section  with 
major  diameters  double  the  minor  diameters.  Many  exact 
measurements  of  these  jets  have  been  made  and  are  partially 
described  in  a  paper  by  Farmer  and  Strickland  in  the  Trans. 
Can.  Roy.  Soc,,  vol.  IV.  sec.  3. 

With  a  semicircular  orifice  the  section  has  a  more  or  less 
semicircular  boundary  and  a  single  sheet  at  right  angles  to  the 
diameter. 

The  common  explanation  of  this  phenomenon  is  that  the 
fluid  particles  issuing  along  different  parabolic  stream-lines 
impinge  upon  each  other,  and  by  their  mutual  reactions  cause 
the  jet  to  spread  out  and  assume  sectional  forms  depending 
upon  the  shape  of  the  orifice. 

Thus  the  fluid  particles  issuing  horizontally  and  freely  at  J3f 
with  a  velocity  \f2gAB,  describe 
a    parabola    BD.       The     particle 
issuing     at     C    with     a    "velocity 
describe  a  parabola  CD 


of  less  curvature  than  BD.  The 
particles  cannot  pass  simultane- 
ously through  the  point  D  and 
must  necessarily  press  upon  each 
other.  They  are  therefore  com- 
pelled to  move  out  of  their  natural 
paths,  and  the  jet  spreads  into 
sheets. 

A  theory  which  seems    more  FlG-  35- 

fully  to  account  for  the  whole  of  the  facts  is  that  the  peculiar 
changes  in  form  are  really  due  to  surface  tension  and  to  the 
differences  between  the  atmospheric  pressure  and  the  internal 
pressure  of  the  jet. 

In   the   case,    for   example,    of  a  jet   flowing    through   an 
elliptical  orifice  with  the  major  axis  vertical,  the  stream-lines. 


50  TIME  OF  FILLING   A   LOCK. 

in  the  vein  are  convergent  and  mutually  react  upon  each  other, 
causing  the  jet  to  contract  vertically  and  elongate  horizontally 
at  a  rate  gradually  increasing  to  a  maximum,  when  the  section 
is  a  circle  in  form. 

At  this  stage  'the  rates  of  elongation  and  contraction  are 
the  same.  The  elongation  and  contraction  still  continue,  but 
at  a  diminishing  rate,  until  the  movement  is  stopped  by  the 
effect  of  surface  tension,  when  the  section  is  again  elliptical, 
with  the  major  axis  horizontal  and  the  minor  axis  vertical. 
The  new  major  and  minor  axes  then  again  begin  respectively 
to  contract  and  to  elongate,  the  section  of  the  jet  passing 
through  the  circular  form  to  its  initial  elliptical  form. 

This  process  is  repeated  over  the  whole  length  of  the 
unbroken  jet,  and,  in  fact,  in  this  portion  of  the  jet  the  surface 
tension  produces  an  effect  similar  to  that  which  would  be  pro- 
duced if  the  jet  were  surrounded  by  an  elastic  envelope. 

If  the  orifice  is  small  and  the  head  not  large,  the  jet,  on 
leaving  the- contracted  section  at  the  orifice,  spreads  out  into 
sheets  and  then  diminishes  to  a  contracted  section  similar  to 
the  first,  after  which  it  again  spreads  out  into  sheets,  bisecting 
the  angles  between  the  first  set  of  sheets,  and  again  diminishes 
to  a  contracted  section.  This  action  is  repeated  so  long  as 
the  jet  remains  unbroken.  A  comparatively  few  experiments 
made  in  the  laboratory  indicate  that  if  the  head  h  is  not  large, 

the  wave-length  oc  vV/  oc  7-. 

15.  Emptying  and  Filling  a  Canal  Lock. — When  the 
liead  varies,  as  in  filling  or  emptying  a  reservoir  or  a  lock,  in 
filling  a  vessel  by  means  of  an  orifice  -under  water,  or  in 
emptying  water  out  of  a  vessel  through  a  spout,  Torricelli's 
theorem  is  still  employed. 

If  the  lock  or  vessel  is  to  be  filled,  Fig.  36,  let  X  sq.  ft. 
be  the  area  of  the  water-surface  when  it  is  x  ft.  below  the  sur- 
face of  the  outside  water. 


EXAMPLES. 


If  the  lock  or  vessel  is  to  be  emptied,  Fig.  37,  then  JTsq. 
ft.  is  the  area  of  the  water-surface  when  it  is  x  ft.  above  the 
orifice. 

In  each  case  x  ft.  is  the  effective  head  over  the  orifice,  and 
is  the  head  under  which  the  flow  takes  place. 


FIG.  36. 


FIG.  37. 


In  the  time  dt  the  water-surface  in  the  lock  or  vessel  will 
ise  or  fall  by  an  amount  dx*.     Then 

—  X .  dx  —  quantity  which  has  entered  the  lock 
=  cA  \'2g* .  dt, 


Xdx 


A  being  the  area  of  the  orifice. 
Hence 

t  = 


CA  1/2gX 

an  equation  giving  the  time  of  filling  or  emptying  the  lock 
between  the  level  x  and  h.  The  value  of  c  for  submerged 
orifices  seems  to  be  somewhat  less  than  when  the  flow  occurs 
freely,  but  it  is  usual  to  take  .6  or  .625  as  a  mean  value. 

Ex.  i.  A  paraboloidal  vessel  with  a  latus-rectum  of  i  ft.  and  5  ft.  in 
height,  is  immersed  in  water  to  a  depth  of  4  feet.  How  long  will  it  take 
to  fill  the  Vessel  to  the  level  of  the  outside  surface  through  an  orifice 
i  inch  in  diar.  at  the  vertex  ?  (Take  c  =  f .) 

Let  2y  ft.  be  the  diar.  of  the  free  surface  when  it  is  x  ft.  above  the 
orifice.  Then 


52  EXAMPLES. 

Also,  if  the  water  rises  dx  ft.  in  dt  sees., 

Tty*dx  =  amount  entering  vessel  in  dt  sees. 

=  quantity  flowing  through  orifice  under  a  head  of 
4  —  x  ft.  in  dt  sees. 


57r-(4  — 


'    576 
and  therefore 

y .  dx  =  xdx  =  -^-7(4  —  x)dt, 
or 


576         .r  576  j  4  -  (4  —  x)  \ 

dt  =  --  -ax  =  -•—  \  -  -  \  dx 

5     (4  -  *)*  5      I      (4  -  *)*      \ 


Integrating   between    the    limits   ,r  =  o     and     x  =  4  ft.,    the    required 
time  in  sees. 


=    1228.8. 

Ex.  2.  The  horizontal  section  of  a  lock-chamber  is  approximately  a 
rectangle  and  its  length  is  360  feet.  The  side  walls  have  a  batter  of  i  in 
12,  and  the  width  of  the  free  surface  when  the  lock  is  full  of  water  is, 
45  feet.  How  long  will  it  take  to  empty  the  lock  through  two  sluices  in 
the  gates,  each  8  ft.  by  2  ft.,  the  sluice  horizontal  centre-line  being  13  ft. 
below  the  free  surface  in  the  lock  and  4  ft.  below  that  of  the  canal  on  the 
down-stream  side  ? 

Let  the  level  of  the  water  in  the  lock  fall  x  ft.  in  /  seconds. 

The  area  of  the  water-surface  is  then 


If  the  level  now  sinks  dx  ft.  in  dt  sees., 

360(45  —  £}dx  =  arnount  of  water  which  has  flowed  out  through  thet 
'  sluices 


=  2  .  |  .  2  .  8  .  f/2  .  32  .  X  .  dt 

=  \6ox*  ,<tt. 


GENERAL   EQUATIONS.  53 

Therefore 


Integrating  between  the  limits  x  —  o  and  or  =  9  ft.,  the  required  time 
in  sees.  =  |- 

=  6oo|. 

16.  General  Equations.  —  Bernouilli's  theorem  may  be 
'easily  deduced  from  the  general  equations  of  fluid  motion,  as 
follows:  — 

Let  /  be  the  pressure  and  p  the  density  at  any  point  whose 
co-ordinates  parallel  to  the  axes  are  x,  y,  z. 

Let  u,  v,  w  be  the  velocities  of  flow  at  the  same  point 
parallel  to  the  axes,  and  let  X,  Y,  Z  be  the  accelerating 
forces.  Then  three  equations  result  from  the  principle  of  the 
^equality  of  pressure  in  all  directions,  viz.  : 

I  dp  d(u]  du  du          du  du 

,     =  X  —  —j—  =  X  --  -r  —  u  -j  --  v-j  --  w  -j-;   (i) 
p  dx  dt  dt          dx          dy  dz     ^  ' 

i  dp         __       d(v\  dv          dv  dv  dv 

---  -=Y--U-V-W 


I  dp  d(w)  dw          dw         dw  dw 

~T~  —  Z  --  -j-  =  Z  --  j  --  u  -j-  —  v  -7~  —  w  -7-.   (3\ 
pdz  dt  dt  dx          dy  dz     u; 

If  the  motion  is  steady,  so  that  the  velocity  at  any  point  is 

du  dv        dw 

a.  function  of  the  position  only,  then  -.-  =  o  —  -y-  =  —  ,  and 

u,  v,  w  may  be  expressed  as  the  differential  coefficients  of  a 
function  F.     Thus, 

dF  dF  dF 

u  =  -j-;     v  =  —  j—  ;     w  —  —j-\ 
dx  dy  dz 


54  GENERAL  EQUATIONS. 

and  therefore 

du  _     d*F_    _  dv_9 
dy  "     dydx   ~~  dx' 

du        d*F        dw 

dz  ~~  dzdx  .     dx  ' 

dv        d*F        dw 
dz       dzdy        dy ' 

Hence  equations  I,  2,  and  3  may  be  written 

I  dp  du  dv  dw 

~r~  =  X  —  u-} v -j w  -7— ;      .     .     .     (4^ 

p  dx  dx  dx  dx 

i  dp  du  dv  dw 

-  -j-  —  Y—  u-j v  --r-  —  w  -T-;      .     .     .     (c) 

p  dy  dy  dy  dy' 

I  dp  du  dv  dw 

— T"  =  ^  —  u-j v  —r  —  w  -j-.     ...     (6) 

p  dz  dz  dz  dz 

Multiplying  eq.  (4)  by  dx,  eq.  (5)  by  dy,  and  eq.  (6)  by 
dz,  and  adding,  then 


*  dx  dy  dz( 

which  may  be  written 

//^ 

-  v  dv  -f-  w  dw). 


P 
Integrating,  and  assuming  the  fluid  to  be  homogeneous, 

P-  =  f(Xdx  +  Ydy  +  Zdz)  -  u'  +  ^  +  w  +  a  constant^ 


LOSS  IN  SHOCK. 


55 


Hence,  if  gravity  is  the  only  force,  and  if  F  is  the  resultant 
.velocity  at  the  point, 


and  the  last  equation  becomes 


Js [-a  constant 

"2 


and  therefore 


=  —  gz  —  ' —  +  a  constant ; 


p         V1 
z  -\ -I =  a  constant. 


17.  Loss  of  Energy  in  Shock. — An  abrupt  change  of  sec- 
tion at  any  point  in  a  length  of  piping  destroys  the  parallelism 
of  the  fluid  filaments,  breaks  up  the  fluid,  and  energy  is  dissi- 
pated in  the  production  of  eddy  and  other  motions.  The 
energy  thus  wasted  is  termed  ' '  energy  lost  in  shock. ' ' 


FIG.  38. 

In  a  short  length  of  piping,  Fig.  38,  where  the  section 
suddenly  changes  from  -A'B'  to  EF,  consider  the  fluid  mass 
between  the  two  transverse  sections  AB,  where  the  motion  of 


56  LOSS  IN  SHOCK. 

the  fluid  filaments  has  been  undisturbed  and  is  in  parallel  lines, 
and  CD,  where  the  parallelism  has  been  again  re-established. 

In  an  indefinitely  short  interval  of  time  /  let  the  mass  move 
forward'  into  the  position  bounded  by  the  plane  sections  A'B' 
and  CD'. 

Let  alt  vl,  pl  be  the  sectional  area,  velocity  of  flow,  and 

mean  intensity  of  pressure  at  A'B'. 
Let  a.2  ,  ?'2  ,  /2  be  similar  symbols  for  C'D'. 
Let  2l  ,  z2  be  the  elevation  above  datum  of  the  C.  G.s  of 

the  sectional  areas  at  A'B'  and  C'D'. 
Let  h  be  the  vertical  distance  between  the  C.  G.s  of  the 

areas  EF  and  A'B'. 
Let  P  be  the  mean  intensity  of  pressure  over  the  annular 

surface  between  EF  and  A'B'. 

The  resultant  force  acting  in  the  direction  of  motion  upon 
the  mass  of  fluid  under  consideration 

—  component  of  weight  of  mass  in  this  direction 

-{-  pressure  on  A'B' 

•  -f-  pressure  on  annular  surface  between  EF  and  A'B' 

—  pressure  on  C'D' 


=  wa2(s,  -z2-  h}  +  ajp,  -/3), 

assuming  that  P  =  pl  ,  or  that  the  mean  intensity  of  pressure 
is  unchanged  throughout  the  whole  of  the  section  EF. 

The  normal  reaction  of  the  pipe-surface  between  EF  and 
C'D'  has  no  component  in  the  direction  of  motion,  and  fric- 
tional  resistances  are  disregarded. 

Hence  the  impulse  of  the  resultant  force 


=  change  of  momentum  in  the  same  direction  of 
the  fluid  masses  CDD'C  and  ABB'  A',  since 
the  momentum  of  the  mass  between  A'B' 
and  CD  remains  unchanged 


LOSS  IN  SHOCK.  57 


w  w 

=  —  a<,v2  .  vJ  --  a,v,  .  v, 

.<r  g  l  ] 

"W 

=    -tf2<>22   -    *WX 

o  ' 


since,  by  the  condition  of  continuity, 


Dividing    throughout    by    the    factor    wa<£,    the    equation 
becomes 


it 

W  W  g  g 

\vhich  may  be  written  in  the  form 

£+!£  =          ^.4.  A+i  +  feHUtf 

~l       "    W~T~  2g   ~  W^~  2g^  2g 

Now  the  pipes  are  nearly  always  axial,  and  in  such  case 
h  =  o,  so  that  the  last  equation  becomes 


^  _i_     _   • 

1h  1 


2g 


If  there  had  been  no  abrupt  change  of  section,  or  if  the 
change  between  A'B1'  and  CD  had  been  gradual,  then  no 
internal  work  would  have  been  done  in  destroying  the  parallel- 
ism of  the  fluid  filaments,  and  no  energy  wasted.  Therefore, 
by  Bernoulli's  theorem,  the  relation 


,        ,.,      +      + 

"  l  "*"    «/  T   2^-   ~  '  'C'2         W          2g 

would  have  then  held  good. 
(v    —  V  )2 

Thus    —  --  2     ft.-lbs.  of  energy  per  pound  of  fluid  is  the 

Joss  in  shock  between  A'B'  and  CD. 

Experiment  justifies  the  assumption  P  =  pr 


BORDA'S  MOUTHPIECE. 


Ex.  At  a  point  A,  150  ft.  above  datum,  a  line  of  piping  suddenly 
doubles  in  sectional  area.  If  the  velocity  of  flow  in  the  larger  pipe  is 
8  ft.  per  sec.,  and  if  the  pressure  *it  A  is  125  Ibs.  per  sq.  in.,  find  the  pres- 
sure per  sq.  in.  at  B,  8  ft.  above  datum,  the  motion  being  steady. 

The  velocity  of  flow  in  the  smaller  pipe  is  evidently  16  ft.  per  second 
Therefore  the  loss  of  head  in  shock  at  the  sudden  change  of  section 


(16  -  8)* 

2.32 


=  i  ft. 


Hence,  if/  is  the  pressure  per  sq.  in.  at  B, 

p  x  144         8* 

8  +  '-*-?  +  ,64  +  '  =  15° 


x  144       i6» 


or 

and 


,     144 
*'  62* 

/  = 


432, 


-  in- 


18.    Mouthpieces.  —  (a)   Borda  s    Mouthpiece. — This    is 
merely  a  short  pipe  projecting  inwards,  as  in  Fig.  39,  which 


FIG.  39- 

represents  a  jet  flowing  through  a  re-entrant  mouthpiece  of  sec- 
tional area  A,  fixed  in  the  vertical  side  of  a  vessel  of  constant 
horizontal  section  and  containing  water  kept  at  a  constant 
level.  The  mouthpiece  is  sufficiently  long  to  allow  of  the  jet 
springing  clear  from  the  end  EF  without  adhering  to  the  inside 
surface. 


BORDJ'S  MOUTHPIECE.  59 

The  velocity  of  the  fluid  molecules  along  AC  and  DK,  is 
sufficiently  small  to  be  disregarded,  so  that  the  pressure  over 
this  portion  of  the  vessel  is  distributed  in  accordance  with  the 
hydrostatic  law.  The  same  may  also  be  said  of  the  pressure 
on  the  remainder  of  the  vessel's  surface. 

Again,  the  only  unbalanced  pressure  is  that  on  the  surface 
HG  immediately  opposite  the  mouthpiece,  and  the  resultant 
horizontal  force  in  the  direction  of  the  axis  of  the  mouthpiece 

=  (A  +  WJi)A  ~  PV&  =  ™kA> 

k  being  the  depth  of  the  axis  below  the  water-surface  and  /0 
the  intensity  of  the  atmospheric  pressure. 

The  jet  converges  to  a  minimum,  or  contracted  section  MN, 
of  area  a. 

In  a  unit  of  time  let  the  fluid  mass  between  AB  and  MN* 
take  up  the  position  bounded  by  A  'B  and  M'N'.  Then 

wkA  =  impulse  of  force  in  direction  of  motion 

=  change  of  momentum  in  same  direction  in  a  unit 

of  time 
=  difference  between  the  momenta  of  MNN'M'  and 

ABB'A\    since   the    momentum    of  the    mass 

between  A'B'  and  MN  remains  unchanged 
=  momentum  of  MNN'M',  since  the  momentum  of 

ABBA'  is  vertical 

w 

=  — <?r  .  r  =   - 
&  ^ 

v  being  the  mean  velocity  of  flow  across  the  contracted  section. 
Hence 

«.•  w 

wkA  =  — «re*  =  —a  . 

«a»  O 

and  therefore 

A  =  2a. 


6o  BORDA'S  MOUTHPIECE. 

or 

1  a 

—  =  —r  =  coefficient  of  contraction. 

2  ^4 

This  result  has  been  very  closely  verified  by  experiment, 
the  coefficient  having  been  found  to  be  .5149  by  Borda,  .5547 
by  Bidone,  and  .5324  by  Weisbach. 

Borda's  mouthpiece  gives  a  smaller  discharge  than  a  sharp- 
edged  orifice,  but  a  discharge  which  is  much  more  uniform, 
and  hence  it  is  generally  used  in  vessels  from  which  water  is 
to  be  distributed  by  measure. 

NOTE. — Let  Fig.  40  represent  a  jet  flowing  through  a 
re-entrant  mouthpiece  of  sectional  area  A,  fixed  in  the  sloping 
-side  of  a  reservoir  containing  water  kept  at  a  constant  level, 
and  suppose  that  the  reservoir  is  of  such  size  that  GHKL  may 
represent  a  cylindrical  fluid  mass,  coaxial  with  the  mouthpiece 
and  so  large  that  the  velocity  at  its  surface  is  sensibly  nil. 
Let  h' ,  h  be  the  depths  below  the  water-surface  of  the  C.  G.s 
of  the  areas  GH  and  KL,  respectively. 


FIG.  40. 

Then  the  resultant  force  along  the  axis  of  the  mouthpiece 
=  pressure  on  GH  —  pressure  on  CK  and  on  DL 


RING-NOZZLE.  61 

—  pressure  on  EF 

-\-  component    of  the   weight   of  the   fluid 
mass  GHKL 

=  (/0  +  wk1)  area  GH  —  (/0  +  wk)  (area  CK-\-  area  Z>Z) 

-  /0  .  area  J^F  +  w  .  area  677.  GK  .     '„  ,  very  nearly 

=  whA. 

Hence,  in  a  unit  of  time, 

whA  =  impulse  of  this  force 

=  change  of  momentum  in  direction  of  axis 

w  w  w 

=  —  av  .  v  =  —  air  =  --  a  .  2gh, 


a  being  the  area  of  the  contracted  section,  while  //  is  also  very 
approximately  the  depth  of  its  C.  G.  below  the  water-surface. 
Thus,  as  before, 

the  coefficient  of  contraction  =  —  ;-  =  —  . 

A         2 

(&)  Ring-nozzle.  —  The  ring-nozzle    (see  Fig.   41)  is  often 

used  with  a  fire-engine  jet,  and 
consists  of  a  re-entrant  pipe  of 
sectional  area  al  fixed  in  a  pipe 
of  sectional  area  ar  The 
length  of  the  re-entrant  portion 
is  such  that  the  water  springs 
clear  from  the  inner  end  and, 
without  again  touching  the 
surface  of  the  mouthpiece, 
FlG>  41-  converges  to  a  minimum  or 

contracted  section  of  area  a  at  MN. 

Consider  the  fluid  mass  between  MN  and  a  transverse  sec- 

tion AB,  and  in  a  unit  of  time  let  it  move  into  the  position 

bounded  by  the  planes  ~M'N'  and  A  '  B'  . 


62  RING-NOZZLE. 

It  is  assumed  that  the  motion  is  steady  and  that  there  is 
no  internal  work  due  to  the  production  of  eddies  or  other 
motions. 

Let/>0,  v  be  the  intensity  of  the  atmospheric  pressure  and 

the  velocity  at  MN. 
Let  pl  ,  7>j  be  the  mean  intensity  of  pressure  and  the  velocity 

at  AB. 
Let    P    be    the    mean    intensity  of  the  pressure  over   the 

annular  surface  EF,  GH. 
Let  ,8*0,  sl  be  the  elevations  above  datum  of  the  C.-G.s  of 

the  sections  MN  and  AB. 
Then 


—  impulse  -in  direction  of  motion 

—  change  of  momentum  in  same  direction  in  a  unit  of  time 

=  difference  of  the  momenta  of  the  fluid  masses  MNN'M'  and 
ABB'  A' 


<? 

Assuming  that  P  =  px ,  the  last  equation  becomes 
By  Bernoulli's  theorem, 

,i  +  A  f  i£  =  ,o  +  £  +  Jl 

and  therefore 

A  ~  A  ^_  ^  -  ^i2  (2) 

Now  ^  —  ^0  is  very  small  and  maybe  disregarded  without 
sensible  error,  and  then,  by  eqs.  (i)  and  (2), 

1  •*   1  .*   0  2     1 


CYLINDRICAL   MOUTHPIECE. 


Hence 


(a a*  — 


If  the  sectional  area  at  of  the  pipe  is  very  large  as  com- 
pared with  a,  so  that  --  may  be  disregarded  without  sensible 

2  I 

error,  then       =     ,  and  therefore  the  coefficient  of  contraction 
a^       a 

=  —  =  — ,  as  in  Borda's  mouthpiece. 
a,        2' 

(c)    Cylindrical  MoutJipicce. — When    water  issues   from    a 

cylindrical  mouthpiece  (see  Fig. 
42)  at  least  two  to  two  and  one- 
half  diameters  in  length,  the  jet 
issues  full  bore,  or  without  con- 
traction at  the  point  of  discharge. 
If  A  be  the  sectional  area  of 
the  mouthpiece,  //  the  depth  of 
its  axis  below  the  water-surface, 
and  Q  the  amount  of  the  dis- 
charge, then  experiment  shows 
that 

0=  .82 A  i/2gh.      .      (i) 

The    coefficient    .82     is    the 
FlG  42  product    of    the     coefficients     of 

velocity  and  contraction,  but  the 

coefficient  of  contraction  is  unity,  and  therefore  the  coefficient 
of  velocity  is  .82.  Now  the  mean  coefficient  of  velocity  in  the 
case  of  a  simple  sharp-edged  orifice  is  .947,  and  the  difference 
between  .947  and  .82  cannot  be  wholly  accounted  for  by  fric- 
tional  resistances,  but  is  in  part  due  to  a  loss  of  head.  In  fact, 
the  water,  as  it  clears  the  inner  edge  of  the  mouthpiece,  con- 


64  CYLINDRICAL   MOUTHPIECE. 

verges  to  a  minimum  section  MN,  of  area  a,  and  then  swells 
out  until  at  M'N'  it  again  fills  the  mouthpiece. 

Energy  is  wasted  in  eddy  motions  between  MN  and  M'N\ 
where  the  action  is  similar  to  that  which  occurs  at  an  abrupt 
change  of  section. 

Let  /,  v  be  the  intensity  of  the  pressure  and  the  mean 
velocity  of  flow  at  the  point  of  discharge. 

Let  pl ,  z/j  be  similar  symbols  for  the  contracted  section  MN. 

Let  /0  be  the  intensity  of  the  atmospheric  pressure. 

(v,  -  vf 
Remembering  that  — is  the  loss  of  head  * '  due  ta 

shock  "  between  MN  and  M'N' ,  then,  by  Bernoulli's  theorem, 

^    I    ..-f  °    __    -*V     i     L_  _   .          [     i     ^   i  '  /2\ 

W    ~       W  2g    '        W  2g  2g 

Hence 

A  -/ 


=  -±-  -  h, (3\ 

w  ^<r  V0> 

h   +     *         '    * 


W  2g 

and 


w  2g 


a          v 
where  c  =  coefficient  of  contraction  =  —  =  — .      Therefore: 

A '        it, 

Po  - 


*>=    •    ,T  w.; (4) 


an  equation  giving  the  velocity  of  flow  at  the  point  of  discharge^ 


CYLINDRICAL   MOUTHPIECE.  65 

If  the  discharge  is  into  the  atmosphere,  p0  =  p  and  equa- 
tion (4)  becomes 


V2  =    -       -        --  rj  =  C.'  .  2gh,        ...      (5) 

i 

where 


If  C,,  =  .62,  then  cv  =  .85,  while  experiment  gives  .82  as 
the  value  of  Cv.  The  small  difference  between  .85  and  .82  is 
probably  due  to  frictional  resistance.  The  value  ..82  for  cv 
makes  C.  approximately  .617. 

Again,  the  discharge  from  a  simple  sharp-edged  orifice  of 
same  sectional  area  as  the  mouthpiece  is  .62  A  \/2gh,  or  more 
than  24  per  cent  less  than  the  discharge  from  the  cylindrical 
mouthpiece. 

The  loss  of  head  between  MNund  M'N' 


? (r  *) &"  \r 

^<s  z&    X6c 

=  W-l  -  i)  (by  eqs.  (5)  and  (6) 

\cv 

=  A(i  -  C*)  =  h  X  .3276  =  .487  X  ^~ 

h 
=  — ,  approximately. 

Thus  the  effective  head  is  only  f  h,  instead  of  h. 

By  eq.  (3),  the  difference  between  the  pressure-heads  at  MN 
and  at  the  point  of  discharge 

*  =  —  -  h   ' 


W  2g 


=  J^,  very  nearly. 


66 


DIVERGENT  MOUTHPIECE. 


Now  if  one  end  of  a  tube  is  inserted  in  the  mouthpiece  at 
the  contracted  section  (Fig.  42)  and  the  other  end  immersed 
in  a  vessel  of  water,  the  water 
will  at  once  rise  to  a  height  kl 
in  the  tube,  showing  that  the  :: 
pressure  at  the  contracted  sec- 
tion is  less  than  that  due  to  the 
atmosphere.  By  careful  meas- 


urement it  is  found  that  //L  is 
very  nearly  equal  to  f//,  which 
verifies  the  theory. 

(d}   Divergent  Mouthpiece. 
— Suppose  that  for  the  cylin-  FIG.  43. 

drical  mouthpiece  in  (c)  there  is  substituted  a  divergent  mouth- 
piece of  the  exact  form  of  the  issuing  jet,  Fig.  43.      Then 

(1)  The  mouthpiece  will  run  full  bore. 

(2)  There  will  'be  no  loss   of  head   between  the  minimum 
section  MN  and  the  plane  of  discharge  AB,  as  there  is  now 
no  abrupt  change  of  section. 

Hence   by  Bernouilli's   theorem,   and  retaining   the   same 
symbols  as  in  (<:), 

i*Ji-h  -=^  +  ^  =  £-  +—  (i) 

W  W  2g "        W  2g' 

If  the  discharge  is  into  the  atmosphere,  p  =/>0,  and  therefore 

(2) 


or,   introducing  a  coefficient  cv  (=  .98,    nearly,  for  a  smooth 
well-formed  mouthpiece), 


,      ......      (3) 


and  the  discharge  is 


=  cvA 


(4) 


DIVERGENT  MOUTHPIECE.  67 

From  the  last  equation  it  would  appear  as  if  the  discharge 
Avt>uld  increase  indefinitely  with  A,  but  this  is  manifestly  im- 
possible. 

In  fact,  by  eq.  (i),  the  flow  being  into  the  air,  and  taking 


W  W 


since  av^  =  Av.      But/L  cannot  be  negative,  and  therefore 


so  that 

A  /1T~ 

wh  +  i   ......     (7) 


A      / 

a       V 


gives  a  maximum  limit  for  the  ratio  of  A  to  a. 

Now  --  =  34  ft.  very  nearly,  and  the  last  equation  may  be 

\V^ 

written 


By  eqs.  (4)  and  (7), 

0=c,ay/2g(h  +  ^))      ...     (9) 

which   is   also  the   expression   for  the  discharge  through   the 
minimum  section  a  into  a  vacuum. 

If,  however,  the  sectional  areas  of  the  mouthpiece  at  the 
point  of  discharge  and  at  the  throat  are  in  the  ratio  of  A  to  a, 
as  given  by  eq.  (7),  it  is  found  that  the  full-bore  flow  will 
be  interrupted  either  by  the  disengagement  of  air,  or  by  any 
slight  disturbance,  as,  for  example,  a  slight  blow  on  the 


68  CONVERGENT  MOUTHPIECE. 

mouthpiece,  and  hence,  in  practice,  it  is  usual  to  make  the 
ratio  of  A  to  a  sensibly  less  than  that  given  by  eq.  (7). 

(e)  Convergent  Mouthpiece. — With  a  convergent  mouthpiece 
(Fig.  44)  two  points  are  to  be  noted : 

(i)  There  is  a  contraction  within  the  mouthpiece,  followed 
by  a  swelling  out  of  the  jet  until  it  again  fills  the  mouthpiece. 


FIG.  44. 

Thus,  as  in  the  case  of  cylindrical  mouthpieces,  there  is  a 
' '  loss  of  head  ' '  between  the  contracted  section  and  the  point 
of  discharge,  and  also  a  consequent  diminution  in  the  velocity 
of  discharge. 

(2)  There  is  a  second  contraction  outside  the  mouthpiece 
due  to  the  convergence  of  the  fluid  filaments.  The  mean 
velocity  of  flow  (v1)  across  the  section  is 


Cy    being  the  coefficient  of  velocity  and  h  the  effective  head 
above  the  centre  of  the  section. 
Also,  the  area  of  this  section 

==  cc'  X  area  of  mouthpiece  at  point  of  discharge 

ct.A, 

ccf  being  the  coefficient  of  contraction.      Hence  the  discharge 
Q  is  given  by 


Q  =  cv'cc'A  V^rk  =  C'A 
'(=  cv'cc')  being  the  coefficient  of  discharge. 


ENERGY  AND  MOMENTUM  OF  JET. 


69 


The  coefficients  cvf  and  c  depend  upon  the  angle  of  con- 
vergence, and  Castel  found  that  a  convergence  of  13°  24'  gave 
a  maximum  discharge  through  a  mouthpiece  2-6  diameters  in 
length,  the  smallest  diameter  being  .05085  foot. 

TABLE   GIVING   CASTEL'S    RESULTS. 


Angles  of 
Convergence. 

C  ' 
c 

C  ' 

V 

C' 

Angles  of 
Convergence. 

Cc 

<v 

C' 

0°      0' 

•999 

.830 

.829 

I3°    24' 

.983 

.962 

.946, 

I      36 

1.  000 

.866 

.866 

14     28 

•979 

.966 

.941 

3    10 

I.OOI 

.894 

•895 

16    36 

.969 

.971 

•938 

4    10 

1.002 

.910 

.912 

19    28 

•953 

.970 

.924 

5    26 

1.004 

.920 

.924 

21        O 

•945 

.971 

.918 

7    52 

.998 

•931 

.929 

23      o 

•  937 

•974 

•913 

8    58 

.992 

•942 

•934 

29     58 

.919 

•  975 

.896 

10     20 

.987 

-950 

•938 

4O     2O 

.887 

.980 

.869 

12        4 

.986 

•955 

.942 

48      50 

.861 

.984 

.847 

19.  Energy  and   Momentum  of   a  Jet (a)  Jet  from  a 

sharp-edge  orifice. 

v* 
The  energy  of  the  jet  =  wav  —  ft.-lbs.  per  second 


ft.-lbs.  per  second 
cj  ft.-lbs.  per  second 


wavhc* 

-  h.  p.  (horse-power) 
550  ' 

pave* 
550 


h.  p., 


/(=  wfi)  being  the  hydrostatic  pressure  due  to  the  head  h,  and 
the  average  value  of  cv  being  .62. 


w 


The  momentum  of  the  jet  =  —  av  .  v  =  wa  —  =  2wahc* 

g  g 


and  this  is  equal  to  the  pressure  in  pounds  produced  by  the  jet 
against  a  fixed  plane  perpendicular  to  its  direction.      Neglect- 


70  EXAMPLES. 

ing  cj*,  the  thrust  is  double  the  hydrostatic  pressure  due  to  the 
head  h. 

(b)  Jet  front  a  Cylindrical  Mouthpiece. 

v2 
The  energy  of  the  jet  =  wAv~  ft.-lbs.  per  second 


ft.  Ibs.  per  second 

ft.-lbs.  per  second 


the  average  value  of  cv  being  .82. 

w  ifi 

The  momentum  of  the  jet  =  —  Av  .  v  =  wA  —  =  2wAhc*. 

£  g 

Ex.  i.  Water  flows  through  a  Borda  mouthpiece  of  A  sq.  ft.  sec- 
tional area  under  a  head  of  h  feet.  If  the  jet  springs  clear  from  the  inner 
edge,  the  discharge  is  29.2$  less  and  the  jet's  energy  41.4  %  greater  than 
when  the  mouthpiece  runs  full. 

Let  -v  be  the  mean  velocity  of  flow  across  the  contracted  section  MN\ 
u  be  the  mean  velocity  of  flow  at  the  mouth  CD  when  the  mouth- 

piece runs  full.     Then  v  =  2u. 
Let  Qi,  Ei  be  the  discharge  and  energy  of  the  jet  when  it  springs 

clear; 

£a,  EI  be  the  discharge  and  energy  of  the  jet  when  the  mouth- 
piece runs  full.     Then 


tuQi  v*       "wA  u3 
and  £,  =..  —  --_-.  —  -. 

When   the  mouthpiece  runs  full,  the  loss  of 
head  between  .#/Arand  CD 


FIG.  45- 

A              A       u*      u* 
Hence  h  4-  ^  =  0+^-^4 1 . 

TV  IV  2&          1g 


EXAMPLES.  71 


^ 

Therefore  Q*  =  Au  =  — -v, 

wQi  if      TV  A       v3 
and  At  =  -       —  = .  — 

^2  g        4|/2' 

<2i        1/2                         (2»  —  <2i 
Hence  -^-  =    -^-  =  .707     and      ^— ^ =  .292. 

£".       41/2  £1  -  £"* 

1  -jr  -  =  1.414   and      —p —     =  •4I4- 

Ex.  2.  Determine  the  discharges  and  energies  of  a  jet  under  a  head  of 
ioo  ft.,  issuing  from  a  6-in.  mouthpiece  which  is  (a)  cylindrical,  (6)  di- 
vergent (bell-mouth),  (c)  convergent,  the  angle  of  convergence  being 
29058'. 


(a)     v  =  .82  1/64  .  100  =  65.6  ft.  per  sec. 

Q  =  —  —  (  —  j    x  65.6  =  i2f|cu.  ft.  persec.  =8o||gals.  per  sec. 


Energy  =  -  =  54.,^  ft.-,bs.  =  98?JJ  H.  P. 


v  =  .98  1/64.  ioo  =  78.4  ft.  per  sec. 

Q  =  -  -  }  x  7^4  =15.4  cu.  ft.  per  sec.  =  96^  gals,  per  sec. 

7    4\  I2  / 

Energy  =  ^  *  '5l  ^  =  ^^  ft..,bs.  =  .^  H.  p. 


(<:)     t/  =  .896  1/64.  ioo  =  71.68  ft.  per  sec.     (See  Castel's  Table.) 

22  i  /6\a 
Q  = !  —1  x  71.68  =  14.08  cu.  ft.  per  sec.  =  88  gals,  per  sec. 

62\  x  1 4^  (7I.68)2 
Energy  =  ^  ~^~  =  70647.807  ft.-lbs.  =  128.45  H.  P. 

Ex.  3.  There  is  a  36-ft.  head  of  water  over  the  2-in.  throat  of  a  bell- 
mouth.  Find  the  greatest  diameter  of  the  mouth  when  open  to  the 
atmosphere  and  running  full,  the  height  of  the  water-barometer  being 
34  feet. 

Let  p,  v  be  the  pres.  and  vel.  at  the  throat ; 
z>0  be  the  vel.  at  the  mouth. 

Then  /_  +  *'_=  34 +  ^  =  36. 

W  2g  ^    ^    2g  * 

V  *  — 

Therefore  ~  —  2,    or     v0  =  8  |/2  ft.  per  sec. 


72  RADIATING   CURRENT. 

The  velocity  in  the  throat  is  greatest  when  the  pressure,  pi,  is  least, 
i.e.,  when  p*  =  o,  and  then 

v* 
o  H  --  =  36,     or    v  —  48  ft.  per  sec. 

A 

If  D  ins.  is  the  diameter  of  the  mouth,  the  discharge 


144     4  144       4 

n 

=  —  cu.  ft.  per  sec., 
H 

and  D*  =  12.726, 

or  D   =  3.56  ins. 

20.  Radiating  Current.—  As  an  application  of  Bernouilli's 
theorem,  consider  the  steady  plane  motion  of  a  body  of  water 
flowing  radially  between  two  horizontal  planes  a  ft.  apart,  and 
symmetrical  with  respect  to  a  central  axis  (Fig.  46). 

Let  v  ft.  per  second  be  the  velocity  at  the  surface  of  a 
cylinder  of  radius  r  ft.  described  about  the  same  axis.  Then 
the  volume  Q  crossing  the  surface  per  second  is 


Q  =  2nr  .  avy 
and  therefore 

rv  =  -  =  a  constant, 


since  Q  is  constant. 

Thus  v  increases  as  r  diminishes,  and  becomes  infinitely 
great  at  the  axis  ;  but  it  is  evident  that  the  current  must  take 
a  new  course  at  some  finite  distance  from  the  axis. 

If  p  is  the  pressure  at  any  point  of  the  cylindrical  surface 
z  ft.  above  datum,  then,  by  Bernouilli's  theorem, 

p        V*  v* 

z  -I  --  4-  —  =  a  constant  —  //=.-  y  4- 
r  w  ^  2  i 


RADIATING  CURRENT. 


73 


denoting  the  dynamic  head  z  -j by  y.      Hence 


a  constant 


2<f 


and  therefore 


—  y)  =  a  constant, 

FIG.  46. 


JL 


FIG.  47. 

is  an  equation  giving  the  free  surfaces  of  the  pressure  columns 
(Fig.  47).  These  surfaces  are  thus  generated  by  the  revolu- 
tion of  what  is  called  Barlow's  curve. 


74  YORTEX  MOTION 

The  surfaces  of  equal  pressure  are  also  given  by  an  equation 
of  the  same  form. 

21.  Vortex  Motion. — A  vortex  is  a  mass  of  rotating  fluid, 
and  the  vortex  is  termed  free  when  the  motion  is  produced 
naturally  and  under  the  action  of  the  forces  of  weight  and 
pressure  only. 

In  the  radiating  current  already  discussed,  assume  that  the 
direction  of  motion  at  each  point  is  turned  through  a  right 
angle  so  that  the  mass  of  water  will  now  revolve  in  circular 
layers  about  the  central  axis.  Also,  if  there  is  a  slow  radial 
movement,  so  that  fluid  particles  travel  from  one  circular 
stream-line  to  another,  it  is  assumed  that  these  particles  freely 
take  the  velocities  proper  to  the  stream-lines  which  they  join. 
Such  a  motion  is  termed  a  free  circular  vortex. 

The  motion  being  steady  and  horizontal,  the  equation 

z  +  -  4-  —  =  a  constant  =  H  (i) 

W  ~  2g 

holds  good  at  every  point  of  a  circular  stream  of  radius  r. 
Again, 

w  .  d  (z  -[-  — )  =  increment  of  dynamic  pressure  between  two 

consecutive  elementary  stream-lines 
=  deviating  force 
=  centrifugal   force  of  an  element  between  the 

two  stream-lines 

=  --  .  dr. 
gr 

But,  by  eq.  (i), 


\      '    wJ  g 

Hence 


/  p\  W  WV* 

w  .  d\z  +  -  1  =  —  -vdv  =  -  -  .  ar, 
\     '   wl  g  gr 


FREE  SPIRAL    YORTEX.  75 

and  therefore 

dv       dr 

^  +  7  =  °> 

so  that  vr  =  a  constant,  and  v  varies  inversely  as  r,  as  in  the 
case  of  the  radiating  current.  Therefore  the  curves  of  equal 
pressure  will  also  be  the  same  as  in  a  radiating  current. 

Free  Spiral  Vortex. — Suppose  that  the  motion  of  a  mass 
of  water  with  respect  to  an  axis  O  is  of  such  a  character  that 
at  any  point  M,  the  components  of  the  velocity  in  the  direction 
of  OM,  and  perpendicular  to  OM,  are  each  inversely  propor- 
tional to  the  distance  OM  from  O.  The  motion  is  thus  equiva- 
lent to  the  superposition  of  the  motions  in  a  radiating  current 
and  in  a  free  circular  vortex ;  and  if  0  is  the  angle  between 
OM  and  the  direction  of  the  stream-line  at  M,  v  cos  8  and 
v  sin  6  are  each  inversely  proportional  to  OM,  and  therefore  # 
must  be  constant.  Hence  the  stream-lines  must  be  equi- 
angular spirals,  and  the  motion  is  termed  a  free  spiral  vortex. 

This  result  is  of  value  in  the  discussion  of  certain  turbines 
and  centrifugal  pumps.  A  steady  free  surface  in  the  case  of  a 
free  spiral  vortex  is  impossible,  as  the  stream-lines  cross  the 
surfaces  of  equal  presure,  which  are  the  same  as  before. 

Also,  if  pQ ,  r0 ,  7'0  are  the  pressure,  radius,  and  velocity  at 
any  other  point  at  the  same  elevation  z  above  datum,  then 

-  ,  i  ,  fi:!,  +  A  +  !i* 

'     W     '      2g  W  2g" 

and  the  increase  of  pressure-head 


Forced  Vortex. — A  forced  vortex  is  one  in  which  the  law 
of  motion  is  different  from  that  in  a  free  vortex.  The  simplest 
and  most  useful  case  is  that  in  which  all  the  particles  have  ar\ 
equal  angular  velocity,  so  that  the  water  will  revolve  bodily, 


76  FORCED  AND   COMPOUND    VORTICES. 

the  velocity  at  any  point  being  directly  proportional  to  the 
distance  from  the  axis. 
As  before, 

dL  ,  A  =^± 

\    '*~  w)  -    g  r  ' 
But 

v  oc  r  =  oor, 

co  being  the  constant  angular  velocity  of  the  rotating  mass. 
Therefore 


Integrating, 


d\z  +  -  J  =  —  r  .  dr. 

\  I        nntl  fr 


v* 


, 
2  -]  --  —  -    -  +'a  constant  =  ---  1-  a  constant. 

r  W  2g  2g 

Hence,  if  /0  ,  r0,  ^0  are  the  pressure,  radius,  and  velocity 
for  any  second  point  at  the  same  elevation  z  above  datum,  then 


W 


If  the    second    point   is    on   the   axis    of  revolution,    then 
r0  =  o,  and  th.e  last  equation  becomes 

P-P, 


\ 


W  2g 

Thus  the  free  surface  of  the  pressure  columns  is  evidently  a 
paraboloid  of  revolution   with  its  vertex 
at  O,  as  in  Fig.  48. 

A  compound  vortex  is  produced  by 
the  combination  of  a  central  forced  vortex 
with  a  free  circular  vortex,  the  free  sur- 
face being  formed  by  the  revolution  of  a 
Barlow  curve  and  a  parabola. 

For  example,  the  fan  of  a  centrifugal 
pump  draws  the  water  into  a  forced  FIG.  48. 

vortex  and  delivers  it  as  a  free  spiral  vortex  into  a  whirlpool - 
chamber  (Chap.  VIII). 


EXAMPLE. 


77 


In  this  chamber  there  is  thus  a  gain  of  pressure-head,  and 
the  water  is  therefore  enabled  to  rise  to  a  corresponding  addi- 
tional height.  James  Thomson  adopted  the  theory  of  the 
compound  vortex  as  the  principle  of  the  action  of  his  voitex 
turbine. 

Ex.  A  centrifugal  pump  of  2  ft.  interior  and  4  ft.  exterior  diar., 
ma^es  336  revols.  per  minute.  The  water  gradually  fills  up  and  flows 
very  slowly  through  the  wheel  into  a  chamber  of  comparatively  much 
larger  diar.,  from  which  it  passes  away  into  the  discharge-pipe.  The 
pressure  at  the  inlet  may  be  taken  to  be  one  atmosphere,  or  2116  Ibs. 
per  sq.  foot. 

Basing  the  flow  through  the  wheel  upon  the  hypothesis  that  the 
velocity  v  of  any  fluid  particle  is  directly  proportional  to  its  distance  r 


84.35 


FIG.  49. 

from  the  axis  of  rotation,  the  law  connecting  the  pressure  p  and  the 
velocity  v  may  be  expressed  in  the  form  (Ex.  I,  p.  21) 


W  1g 

At  the  inlet/  =  2116,  and  let  v  —  v\.  .  Then 
2116  v? 


so  that 
But 


p  —  2116 


125/336   22    V  T?       r* 

=  — §    ^ 2  )  =  1210,     and     —  —  -  = 

128\    60       7       j  Vi*          I2 


7  8  LARGE   ORIFICES. 

Therefore 

p  =  2116  +  1210  (>'—  i)  =  906  +  i2iora, 

Giving    r,    successively, 

the  values  i,         1.2,          1.4,          1.6,         1.8,      2ft., 

the  corresponding  values 

of  /are  2116,    2648.4,  3277.6,   4003.6,  4826.4,    5746165. 

Thus  the  curve  AB,  obtained  by  plotting  these  values,  shows  the 
variation  of  the  pressure  inside  the  wheel. 

The  hypothesis  of  the  flow  in  the  surrounding  chamber  is  that  the 
velocity  of  any  fluid  particle  is  inversely  proportional  to  its  distance  from 
the  axis  of  rotation  ;  and  in  this  case  the  pressure  and  velocity  are  con- 
nected by  the  relation  (Ex.  i,  p.  21) 

p  v* 

W  2g' 

At  the  wheel  outlet,  i.e.,  where  r  =  2  ft.,/  =  5746  Ibs.  per  sq.  ft.,  and 

let  v  =  v*. 

Then  1746  =  c  _^. 

W  2g* 

therefore  p  =  5746  +  ^(i  -  --} 

2g     \  V**} 


But 


therefore 


125  /3^6  22    \'  7/2 

—4    =  4840,     and      -  =  -> 
1 28  \  60     7   7  T/2       r  ' 


p  =  5746  +  4840(1  -  i}  =  10586  -  -^ 


Giving  r,  successively,  the  values     2,     2.2,   2.4,     2.6,     2.8,     and    3ft., 
the  corresponding  values 

of /are  5746,   6586,  7225,   7723,    8117,    and   8435  Ibs. 

Thus  the  curve  BC,  obtained  by  plotting  these  values,  shows  the  va- 
riation of  the  pressure  in  the  chamber  surrounding  the  wheel. 

22.   Large  Orifices   in  Vertical    Plane    Surfaces. — The 

issuing  jet  is  approximately  of  the  same  sectional  form  as  the 
orifice,  and  the  fluid  filaments  converge  to  a  minimum  section 
as  in  the  case  of  simple  sharp-edged  orifices. 

(a)  Rectangular  Orifice  (Fig.  50). — Let  E,  F  be  the  upper 
and    lower    edges    of  a    large   rectangular    orifice    of  breadth 


LARGE  RECTANGULAR   ORIFICES. 


79 


J?,  and   let  Hl ,  //2  be  the  depths  of  E  and  F,   respectively, 
below  the  free  surface  at  A       If  u  be  the  velocity  with  which 

u* 
the  water  reaches  the  orifice,  then  H  —  — ,  is  the  fall  of  free 

ig 

surface    which    must   have    been    expended    in    producing    the 
velocity  u. 

Hence    H  v  -\-  H  and  7/2  +  H  are  the  true  depths  of  the 
edges  E  and  F  below  the  surface  of  still  water. 

Let    MN  be    the    minimum    or    contracted    section,    and 

assume  that  it  it  is  a  rectangle  of  breadth  b, 
Let  hl ,  //2  be  the  depths  of  M  and  N,  respectively,  below 

the  free  surface  at  A. 
Then   /^  +  //,  hz  +  H  are  the  true   depths   of  M  and  N 

below  the  surface  of  still  water. 

First.      Let  the  flow  be  into  the  air,  the  orifice  being  clear 
above  the  tail-water  level,  Fig.  50. 

Consider  a   lamina  of  the  fluid  at  the   section  MN,  of  the 


4, 


lr 


_.;_.,__  .t_. 


.i_ 


i, 


FIG.  50. 

width   of  the   section,  and  between  the  depths  x  and  x  +  dx 
below  the  surface  of  still  water. 

The  elementary  discharge  dq,  in  this  lamina,  is 


dq  =  bdx  \/2gx, 


8o  LARGE  RECTANGULAR  ORIFICES. 

and  therefore  the  total  discharge  Q  across  the  section  MN  is 

Q 


/fkt  + 
dq  =     I  b  . 
J  hi  + 


dx 

+  H 


e  = 


Then 


IT)*}.. 


The  coefficient  ^  is  by  no  means  constant,  but  is  found  to 
vary  both  with  the  head  of  water  and  also  with  the  dimensions 
of  the  orifice,  and  can  only  be  determined  by  experiment. 

Second.    Let  the  orifice  be  partially  (Fig.  51)  submerged, 
and  let  H "3  be  the  depth  between  the 
surface  of  the  tail-race  water  and  the 
free  surface  at  A. 

By  what  precedes,  the  discharge 
Ql  through  EG,  the  portion  of  the 
orifice  clear  above  the  tail-race,  is 


-(/f  ,  +  #)*}.       (2) 

Every  fluid  filament  flows  through 
the  portion  GF  of  the  orifice  under 
an  effective  head  Hz  +  H,  and  there- 
fore with  a  velocity  equal  to 


FIG.  51. 


Hence  the  discharge  Qt  through  GF  is 


',  ...     (3) 


and  the  total  discharge  Q  is  equal  to  Ql  -\-  Q2. 


LARGE  RECTANGULAR   ORIFICES. 


81 


-,  The  coefficients  cl ,  c2  are  to  be  determined  by  experiment, 

and  if  cl  —  c2  =  c, 


•     (4) 


Third.    Let   the    orifice   be   wholly  submerged   (Fig.    52). 
Then  the  total  discharge  Q  is  evidently 


H,    (5) 

c  being  a  coefficient  to  be  determined 
by  experiment. 

If  the  velocity  of  approach,  u,  is 
sufficiently  small  to  be  disregarded 
without  sensible  error,  then  H  —  o, 
and  equations  (i),  (4),  and  (5),  respec- 
tively, become 


Q=   -cB 

J 


-  H?};       (6) 


(7) 


(8) 


(b)  Circular  Orifices.  —  Let  Fig.  53  represent  the  minimum 
section  of  the  circular  jet  issuing  from  a  circular  orifice. 

Let  26  be  the  angle  subtended  at  the  centre  by  the  fluid 
lamina  between  the  depths  x  and  x  +  dx  below  the  surface  of 
still  water. 

Let  r  be  the  radius  of  the  section  so  that  2r  =  h2  —  h^  ,  h^ 
and  h^  being,  as  in  (a),  the  depths  of  the  highest  and  lowest 
points  of  the  orifice  below  the  free  surface  at  A. 


82 


LARGE  CIRCULAR  ORIFICES. 


H,  as  before,  is  the  head  corresponding  to  the  velocity  of 
approach. 


FIG.  53- 
Then  the  area  of  the  lamina  under  consideration 

=  2r  sin  8  .  dx, 

and  the  elementary  discharge,  dq,  in  this  lamina,  is 
dq  —  2r  sin  0  .  dx 

h^  +  H+h^  +  H  /i, 

But  x  =  -  -J-J  --  r  cos  0  =  -1 


r  cos  0, 


and  therefore 
Hence 


dx  = 


=  2r   sn 


277 


—  r  cos 


and  the  total  discharge  (2  is 


—  r  cos  0 


(9) 


Ex.  The  free  surface  on  the  up-stream  side  is  5  ft.  and  on  the  down- 
stream side  i  ft.  above  the  sill  of  a  rectangular  sluice  12  ft.  wide.  How 
much  must  the  sluice  be  raised  to  give  105,000  gals,  per  minute  ? 

105,000 
105,000  gals.  per.  mm.  =  =  280  cu.  ft.  per  sec. 


NOTCHES  AND   WEIRS.  83 

Let  x  ft.  be  the  opening  above  the  sill.  For  a  depth  of  i  ft.  above 
tne  sill  the  discharge  is  under  a  constant  head  of  5  —  i  =  4  ft.  For  the 
remainder  of  the  opening  the  discharge  takes  place  freely  through  a 
rectangular  orifice,  with  its  upper  and  lower  boundaries  respectively 
<5  —  x}  ft.  and  4  ft.  below  the  up-stream  surface.  Then 


280=.  12.  1. 


.  4*  +7. 

3 


-(5  -  x}\  +  4!} 


=  440  —  40(5  —  x)*. 
Therefore  (5  —  *)*  =  4,     and     x  =  2.48  ft. 

23.  Notches  and  Weirs.  —  When  an  orifice  extends  up  to 
the  free-surface  level  it  becomes  what  is  called  a  notch. 


STILL  WATER  LEVEL 


FIG.  54. 


FIG.  55. 


A  weir  is  a  structure  over  which  the  water  flows,  the  dis- 
charge being"  in  the  same  conditions  as  for  a  notch,  and  is  very 
useful  for  gauging  the  flow  of  small  streams,  the  amount  of 
water  supplied  to  hydraulic  motors,  etc. 

Rectangular  Notch  or  Weir.  —  The  discharge  may  be  found 
by  putting  H1  =  o. 

Thus  equation  (i)  becomes 

Q  =  jc£^i(ffs+.ff)*-ff*}.      .     .     (10) 

If  the  velocity  of  approach  be  disregarded,  then  H  ~  o, 
and  the  last  equation  becomes 


(ii) 


84 


WEIRS. 


and  H2  is  the  depth  to  the  bottom  of  the  notch  or  to  the  crest 
of  the  weir. 

Great  care  should  be  taken  in  obtaining  the  accurate  value 
of  Hz.  A  hook  or  a  stiff  vertical  rod,  with  a  sharp  point,  may 
be  fixed,  at  a  suitable  distance  (5  to  8  ft.)  from  the  back  of 
the  weir,  with  the  point  on  a  level  with  the  crest  of  the  weir. 
The  flume  is  then  filled  with  water  rising  slightly  above  the 
crest  and  producing  a  capillary  elevation  of  the  surface  at  the 
point.  The  water  is  now  allowed  to  subside  until  the  eleva- 
tion is  barely  perceptible,  when  a  hook-gauge  (Chap.  Ill)  is 
adjusted  and  a  reading  taken.  A  second  reading  is  taken  for 
any  required  discharge  over  the  weir,  and  the  difference 
between  the  two  readings  is  the  depth,  //2,  of  the  water  on 
the  crest. 

It  has  been  found  that  the  discharge  (Q)  is  appreciably 
affected  by  vibration,  and  it  is  therefore  of  importance  that  the 
weir  should  be  made  as  rigid  as  possible.  The  up-stream  face 
of  the  weir  is  nearly  always  vertical  and  at  right  angles  to  the 
direction  of  flow. 

To  diminish  the  effect  of  the  velocity  of  approach,  the 
water-section  in  the  flume  should  be 
large  as  compared  with  the  section 
of  the  waterway  on  the  crest,  and 
the  depth  of  the  weir  should  therefore 
be  at  least  twice  the  depth  H 2  of  the 
water  on  the  crest. 

The  crest  should  be  horizontal 
and,  generally  speaking,  it  consists 
of  a  plate  with  a  bevelled  edge,  Fig. 
54,  on  the  up-stream  side,  or  of  a 
thin  plate,  'Fig.  55,  so  that  the  water  springs  clear  from  the 
inner  edge. 

A  rounded  edge,  Fig.  56,  diminishes  the  discharge  and 
should  be  avoided,  as  its  effect  is  uncertain. 


WEIRS. 


The  length  B  of  the  crest  should  be  at  least  three  times  the 
•depth  HY 

The  effective  sectional  area  of  the  water  flowing  through  a 
rectangular  notch,  or  over  a  weir,  is  less  than  BH2 ,  because 
of  (a)  crest  contraction,  (b)  end  contraction,  (c)  the  fall  of  the 
free  surface  towards  the  point  of  discharge. 

It  is  reasonable  to  assume  that  the  diminution  of  the  actual 
sectional  area,  BH2,  due  to  crest  contraction  and  to  the  fall 
of  the  free-surface  level  is  proportional  to  the  width  B  of  the 
opening. 

Suppressed  Weir,  or  Weir  without  End  Contractions. — If 
.a  weir  occupies  the  whole  width  of  the  stream,  or  flume,  Figs. 
57  and  59,  the  contraction  at  each  end  is  wholly  suppressed, 


FIG.  57. 


FIG.  58. 


FIG.  59. 


and  crest  contraction  only  takes  place,  i.e.,  the  falling  sheet 
of  water  is  reduced  in  thickness  near  the  crest.  Air  must  be 
freely  admitted  below  the  falling  sheet,  as  otherwise  a  partial 
or  complete  vacuum  will  be  produced  and  the  sheet  will  be 
depressed  or  will  adhere  to  the  face  of  the  weir,  while  the  dis- 
charge Q  will  be  very  sensibly  modified.  Francis  effected  the 
free  admission  of  air  and  also  prevented  the  lateral  spreading 
of  the  sheet,  after  leaving  the  crest,  by  prolonging  the  upper 
portions  of  the  flume  sides  a  short  distance  beyond  the  weir, 
Fig.  58.  The  discharge  was  thereby  diminished  by  about 
.4  per  cent. 


86 


WEIRS. 


Weir  with  End  Contractions. — These  contractions  occur 
when  the  sides  of  the  weir,  or  notch,  Figs.  60  and  61,  are  at 
a  distance  from  the  sides  of  the  channel,  and  they  have  the 


FIG.  60. 


FIG.  61. 


effect  of  diminishing  the  discharge.  The  contraction  is  com- 
plete, i.e.,  as  great  as  it  can  be  made,  when  the  distance  of  the 
weir  side  from  the  channel  side  is  not  less  than  about  the 
depth  H2. 

Other  things  being  equal,  the  contraction  and  its  effect 
upon  the  discharge  increase  with  H2 .  The  effect  of  end  con- 
tractions is  almost  inappreciable  and  may  be  disregarded  when 
the  length  B  of  the  crest  is  not  less  than  about  Hz  ;  but  as  the 

r> 

ratio    -fj-   diminishes,    the    effect    rapidly    increases.      Francis 

^2 

found  that  the  discharge  for  a  weir  with  perfect  end  contrac- 
tions and  in  which  B  =  4//2 ,  was  diminished  6  per  cent. 

In  his  Lowell  weir  experiments  he  also  found  that,  for 
depths  //"2  -[-  //"over  the  crest  varying  from  3  ins.  to  24  ins., 
and  for  widths  B  not  less  than  three  times  the  depth,  a  per- 
fect end  contraction  had  the  effect  of  diminishing  the  width  of 
the  fluid  section  by  an  amount  approximately  equal  to  one- 


tenth  of  the  depth,  or 


B  - 


10 


so  that  the  effective  width  = 


10 


Thus,  if  there  are  n  end  contractions,  the  effective  width 


—  B  —  —  (H2  -f-  H),   and  the  equation  giving  the   discharge 


WEIRS.  87 

becomes 

.     (12) 

According  to  Francis  the  average  value  of  c  in  this  equa- 
tion is  .622. 


Then  \c  ^2g  =  3.33,  very  nearly,  and  therefore 
Q  =  3-335  -      (//;  +  H)(Ht  +  //)*  -  /tf.  .     (13) 


In  experiments  carried  out  by  Fteley  and  Stearns  with 
suppressed  weirs,  as  described  above,  the  total  variation  in  the 
value  of  the  coefficient  was  found  to  be  about  '2-J-  per  cent. 
The  depths  H2  were  measured  6  ft.  from  the  weir,  and  for  values 
of  H2  exceeding  .07  ft.  they  deduced  the  formula 

(2  =  ^(3.31^2*  +.007), 

in  which  the  velocity  of  approach  is  disregarded. 

Allowance  may  be  made  for  the  velocity  of  approach  by 
substituting  for  H2  the  expression  HI~\-  \\H  according  to 
Fteley  and  Stearns,  but  H2  -\-  i^H  according  to  Hamilton 
Smith,  Jr.,  who  bases  his  conclusions  upon  a  comparison  of 
the  experiments  of  Fteley  and  Stearns  with  those  of  Francis 
and  others. 

f-f  —I—  f-f 

If  the  weir  has  n  end  contractions,  B  —  n  —  ^—^-  —  must  be 

10 

substituted  for  B,  and  allowance  is  made  for  the  velocity  of 
approach  by  substituting  for  H2  the  expression  //"2  -f-  2.05/7, 
according  to  Fteley  and  Stearns,  or  Hz  -f-  1.4/7",  according  to 
Hamilton  Smith,  Jr. 

Hunking  and  Hart  give  the  formula 


88 


SUBMERGED    WEIRS. 


in  which  /*  is  very  nearly  =  I  -| (~^2)  >  where 


S  = 


sectional  area  of  waterway 


10 


Bazin  gives  the  formula 
Q  =  c 

in  which,  if  the  velocity  of  approach  is  disregarded, 

.    .00984 

c  =  .405  +  -rfA 
•"i 

but  if  allowance  is  to  be  made  for  the  velocity  of  approach, 


x  being  the  height  of  the  weir. 

Bazin  considers  that,  with  suppressed  weirs,  as  already 
described  and  which  are  not  very  low,  the  results  obtained 
with  this  coefficient  are  accurate  within  I  per  cent. 

Submerged,  or  Drowned,  Dams  (or  Weirs), — In  these  the 
surface  of  the  tail-race  water  rises  above  the  top  of  the  dam, 


STILCWATER  LEVEL 


FIG.  62. 


Fig.  62.      It  may  be  assumed  that  between  a  and  b  the  flow 
is  as  over  the  crest  of  a  weir,  the  depth  of  water  on  the  crest 


INCLINED   WEIRS. 


«9 


being  //2  -j-  //,  and  that  between  b  and  c  the  flow  is  equiva- 
lent to  that  through  a  submerged  orifice  under  a  constant  head 
H^  _|_  H,  Hence,  if  H'  is  the  depth  of  the  top  of  the  dam 
below  the  surface  of  the  tailwater,  and  if  c  is  the  coefficient  of 
discharge  both  for  the  flow  between  a  and  b  and  also  that 
between  b  and  c, 


Q=-c 

o 


The  following  table  gives  approximate  values  of  c  corre- 

TTf 

sponding   to   different   values   of  the   ratio    -yr 


i 


as 


deduced  from  experiments  carried  out  by  Francis,  the  head 
over  the  crest  varying  from  I  to  2.32  ft.,  and  by  Fteley  and 
Stearns,  the  head  varying  from  .325  to  .815  ft.  : 


Values  of 
H1 


Corresponding4 Values  of  c  as  deduced  from  the  experiments  of 

Fteley  and  Stearns. 


.625  to  .635 


Hi  +  H+H' 
0C  . 

v  rar 
623  t 

LC1S. 

o  .632 

'  .630 
'.625 
•  .615 

1  .610 

'  .607 

'  .607 
•  .607 

1  .607 

620 

20   

610 

an 

•  t;a8 

dO 

586 

en 

egc 

60  

,  .e8q 

.70.  . 

...  .$8$ 

80 

egc 

QO 

.05.  . 

.618 
.600 
.590 

.585 
.583 

.580 

.581 
.590 

.610 


.628 

.610 

.600 

•595 
•593 

•  590 

•  591 
.600 

'  -615 

( Trautwine.} 


Inclined  Weirs. — If  the  up-stream  face  of  a  weir,  instead  of 
being  vertical,  is  inclined  up-stream,  Fig.  63,  the  discharge  is 
diminished,  the  depression  of  the  upper  surface  of  the  falling 
sheet  of  water  commences  near  the  crest,  while  the  lower  sur- 
face rises  higher,  above  the  crest,  and  moves  backwards. 

If  the  face  is  inclined  down-stream,  Fig.  64,  the  discharge 
is  increased,  the  depression  of  the  upper  surface  commences  at 
a  point  farther  from  the  crest  than  when  the  face  is  vertical, 


CIRCULAR  NOTCH. 


while  the  lower  surface  becomes   more  flattened   and   moves 
away  from  the  weir. 

Values  of  the  coefficient  of  discharge  for  inclined  weirs  have 
been  deduced  by  Bazin  and  are  given  in  a  subsequent  article. 


FIG.  63. 


FIG.  64. 


The  discharge  is  increased  by  rounding  the  up-stream  edge 
of  the  weir. 

Circular  Notch. — In  equation  (9),  Art.  22,  put  h^  =  o  and 
hn  =  2r.  Then 


Q  =  2r2 


sin2  0ff+  2r  sin2-)V0, 


and  if  the  velocity  of  approach  be  disregarded,  so  that  H  —  o, 

//i 
sin*  0\  sin  -dO 

r  (2  sin  -  -  sin  ^  +  si 
°     * 


sn 


64 

-- 

15 


Ex.  i.  A  dam  with  a  rectangular  notch  6  ft.  wide  is  formed  across  a 
channel;  and  the  depth  of  the  water  over  the  sill  is  12  ins.  Find  the 
quantity  of  flow  when  the  notch  has  (a)  no  side  contraction  ;  (&)  one 
side  contraction  ;  (?)  two  side  contractions. 


EXAMPLES.  91 

Disregarding  the  velocity  of  approach,  and  assuming  the  coefficient 
of  discharge  to  be  the  same  in  each  case,  viz.,  f, 

(a)  Q1  =  —  .  —•  1/64  .  if  .  6  =  20  cu.  ft.  per  sec. 

0)  fit  =  y  •  J  •  4/64.  Ia'(6  ~  A)  =  1  91  cu.  ft.  per  sec. 

(c)  Qt  =  ~.l.  4/64  .  ii(6  -  A)  =  io£  cu.  ft.  per  sec. 

Ex.  2.  400  cu.  ft.  of  water  per  second  are  conveyed  by  a  channel  of 
rectangular  section  25  ft.  wide,  when  the  water  runs  4  ft.  deep.  Find 
the  height  of  a  dam  built  across  the  channel  which  will  increase  the 
depth  50  per  cent,  taking  into  account  the  velocity  of  approach. 

400  8 

The  velocity  of  approach  =  -  7  =  -  ft.  per  sec. 

/8\2  j 

The  corrresponding  head  =  ~—  =  -  ft. 

Let  x  ft.  be  the  height  of  the  dam. 

First.  Assume  that  the  dam  is  not  drowned,  i.e.,  that  its  crest  rises 
above  the  water-surface  on  the  down-stream  side.  Then 


400  =  y.. 
or  (6  —  x  +  1)1  =  4.837037  =  (6.1  1  1  —  *)§, 

and  x  —  3.25  ft.,  which  is  less  than  4  ft.,  and  therefore  the  assumption 
that  the  dam  is  not  drowned  is  incorrect. 

Second.  Assuming  that  the  dam  is  drowned,  the  discharge  now  takes 
place  under  a  constant  head  of  (2  -f  £)  ft.  for  a  depth  of  (4  —  x}  ft.,  and 
as  over  a  weir  for  a  depth  of  2  ft.  Then 

400  =  |  .  25(4  -  *)  4/6^(2  +  l)*+'y  •{  •  25  .  4/64.  {(2  +  1)1  -  ft)!}, 

or  1.1798  =  (4  —  *)(2l)*» 

and  x  =  3.188  ft.,  which  is  less  than  4  ft.,  and  therefore  the  assumption 
of  a  drowned  dam  is  correct. 

Ex.  3.  If  x  is  the  depth  of  water  over  the  crest  of  a  rectangular 
notch,  then,  disregarding  the  velocity  of  approach, 

Q  = 


9  2 


TRIANGULAR  NOTCH. 


Let  dQ  be  the  change  in  the  discharge  corresponding  to  a  change 
dx  in  the  depth  on  the  sill.     Then 

.  -.r*  .  <ix, 


Hence 


Q 


Thus  a  change  of  6  per  cent  in  the  discharge  corresponds  to  a  change 
of  4  per  cent  in  the  sill  depth,  and  a  change  of  10  per  cent  in  this  depth 
corresponds  to  a  change  of  15  per  cent  in  the  discharge. 

24.  Triangular  Notch.  —  Disregard  the  velocity  of  approach 
and  let  B  be  the  width  of  the  free  surface. 

As  before,   consider  a  lamina  of 
fluid    between    the     depths    x    and         u  -------  -B-  --------  -> 

x  +  dx. 

The  area  of  the  lamina 


FIG.  65. 


and  the  discharge  in  this  lamina  is 


Hence  the  total  discharge  Q  is 


Q  =  c 


=       cB 


cu.  ft.  per  sec. 


(14) 


c  is  a  coefficient  introduced  to  allow  for  contraction,  etc., 
and  Professor  James  Thomson  gives  .617  as  its  mean  value  for 
a  sharp-edged  triangular  notch. 


EXAMPLES.  93 

T) 

%Now  the  ratio  -77-  is  constant  in   a  triangular  notch  and 

2 

varies  in  a  rectangular  notch.  Hence  Thomson  inferred  and 
showed  by  experiment  that  the  value  of  c  is  more  uniform  for 
triangular  than  for  rectangular  notches,  and  therefore  also  the 
former  must  give  more  accurate  results. 

If  the  flow  is  through  a  90°  notch,  B  =  2//2 ,  and 


5 


Q  =  —  c  \/2gH^  =  2.64//"2*  cu.  ft.  per  sec.,  approximately, 

or 

=  158.385/^2^  cu.  ft.  per  min., 

c  being  .617  and  g  =  32.  176. 

Ex.  i.  A  reservoir  discharges  through  a  sharp-edge  triangular  notch, 
and  in/  sees,  the  depth  of  the  water  in  the  notch  Jails  from  H  ft.  to  x  ft. 

Let  S  be  the  sectional  area  of  the  reservoir  corresponding  to  the  x 
ft.  depth  ;  let  mx  be  the  width  of  the  free  surface  on  the  notch  corre- 
sponding to  the  x  ft.  depth,  m  being  a  numerical  coefficient  depending 
up>on  the  notch  angle. 

Then,  since  the  water  sinks  dx  ft.  in  dt  sees., 

—  5.  dx  =  discharge  from  reservoir  in  dl  sees. 

=  amount  flowing  through  notch  in  dt  sees. 
=  —  Vzgc  -  mx*.  •  dt, 


or  dt  =  ---      —  x~    .  dx. 

4 


Hence  the  time  in  sees,  in  which  the  depth  falls  from  H  ft.  to  x  ft. 


4 
If  the  horizontal  sectional  area  6"  is  constant, 

55       /  I  i    \ 

the  time  in  sees.  =  -  —  5  --  -77^  I. 


For  a  90°  notch  m  =  2,  and  taking^  =  32  and  c  =  f, 

SI   \  i    \ 

the  time  in  sec*.  =  --p  _  _J. 


94 


BROAD-CRESTED   WEIRS. 


The  time  becomes  infinite  when  x  =  o,  which  indicates  that  the  flow 
diminishes  indefinitely  with  the  depth  in  the  notch. 

Ex.  2.  Find  the  discharge  in  gallons  per  minute  through  a  90°  sharp- 
edge  notch  when  the  water  runs  4  ft.  deep.  If  the  reservoir  supplying 
the  water  has  a  constant  horizontal  sectionalarea  of  80,000  sq.  ft.,  in 
what  t-ime  will  the  level  sink  3  ft.  ? 

Q  =  ~  ^64  .  |- .  2  .  4s  =  85^  cu.  ft.  per  sec.  =  85*  x  6*  x  60  gals,  per  min. 

=  32,000  gals,  per  min. 

the  time  — ( i >  )  =  17,S°°  secs-  =  4**  hours. 

4V         4*  / 

25.  Broad-crested  Weir.— Let  Fig.  66  represent  a  stream 
flowing  over  a  broad-crested  weir.      On  the  up-stream  side  the 


FIG,  66. 

free  surface  falls  from  A  to  B.  For  a  distance  BD  on  the  crest 
the  fluid  filaments  are  sensibly  rectilinear  and  parallel;  the 
inner  edge  of  the  crest  is  rounded  so  as  to  prevent  crest  con- 
traction. 

Consider  a  filament  ab,  the  point  a  being  taken  in  a  part 
of  the  stream  where  the  velocity  of  flow  is  so  small  that  it  may 
be  disregarded  without  sensible  error. 

Let  A  be  the  thickness  MN  of  the  stream  at  b. 

Let  the  horizontal  plane  through  N  be  the  datum  plane. 

Let  z^ ,  .s-  be  the  depths  below  the  free  surface  of  a  and  b. 

Let  h   be  the  elevation  of  a  above  datum. 


BRO4D-CRESTED   WEIRS.  95 

-,     Let  pQ  ,  p^  ,  /  be  the  atmospheric  pressure  and  the  pressures 
at  a  and  b. 

Let  v  be  the  velocity  of  flow  at  b. 

Then,  by  Bernouilli's  theorem, 


But 

4  =  ^+4     and     ^  =  *+4; 

W  '     2f  W  '     W 

therefore 


IV  10)          2ff 

and  hence 

V* 

2f         i  +  *i-         =X2-^ 

H2  being  the  depth  of  the  crest  of  the  weir  below  the  surface  of 
still  water. 

Thus,  if  B  be  the  width  of  the  weir,  the  discharge  Q  is 


-  A).       ....      (16) 

From  this  equation  it  appears  that  Q  is  nil  both  when 
A  =  o  and  when  A  =  Hv  Hence  there  must  be  some  value 
of  A  between  o  and  H2  for  which  Q  is  a  maximum.  This  value 
may  be  found  by  putting 

dQ  =  «  =  B  ^g(  VH^  -  ~^^, 

and  therefore 


96  BROAD-CRESTED   WEIRS. 

and  the  expression  for  the  discharge  becomes 


Q  =  BH*  S&**  =  -385^     2&ff,        •      (17) 

which  is  the  maxmum  discharge  for  the  given  conditions. 

Experiment  shows  that  the  more  correct  value  for  the  dis- 
charge is 


.    .....      (18) 

If  the  water  approaches  the  weir  with  an  appreciable  velocity 

U* 

u,  corresponding  to  the  head  H,  so  that  —  -  —  //,  then 


and 


This  formula  agrees  with  the  ordinary  expression  for  the 
discharge  over  a  weir  as  given  by  equation  (n),  if  £  =  -525. 

It  might  be  inferred  that  for  broad-crested  weirs  and  large 
masonry  sluice-openings  the  discharge  should  be  determined 
by  means  of  equation  (18)  rather  than  by  the  ordinary  weir 
formula,  viz.,  equation  (n). 

It  must  be  remembered,  however,  that  in  deducing  equa- 
tion (17),  frictional  resistances  have  been  disregarded  and  the 
gratuitous  assumption  has  been  made  that  the  stream  adjusts 
itself  to  a  thickness  /  which  will  give  a  maximum  discharge. 
The  theory  is  therefore  incomplete. 

The  discharge  over  a  sharp-crested  weir  is  sensibly  the 
same  as  that  over  a  weir  with  an  apron,  as  in  Fig.  66,  so  long 
as  the  depth  of  the  water  on  the  crest  is  not  less  than  about 
15  ins.,  but  below  this  limit,  the  discharge  over  the  apron 


RESERVOIR  SLUICES. 


97 


rapidly  diminishes  with  the  depth.  For  example,  the  dis- 
charge over  a  sharp-crested  weir  is  approximately  double  that 
over  a  weir  with  an  apron  when  the  depth  is  about  I  in.,  is 
20  per  cent  greater  when  the  depth  is  6  ins.,  and  10  per  cent 
greater  when  the  depth  is  12  ins. 

26.  Reservoir  Sluices. — The  water  flows  into  the  receiving 
^channel  either  freely,  as  in  Fig.  67,  or  under  water,  as  in 
Fig.  68. 


FIG.  67. 
In  the  first  case,  the  stream-lines  converge  to  a  contracted 


•• 


FIG.  68. 


section,  and  between  the  sluice  and  a  certain  section  DE  there 
is  a  sudden  swell,  the  height  of  swell  being  given  by 


7,  2  7,  2 

EC  —  -i-    -  -* 

J-)  \s      ' 


and  vl ,  ^2  being  the  velocities  of  flow  across  the  contracted 
section  and  the  section  at  DE. 

Let  Alt  A2  be  the  areas  of  the  sluice  and  section  at 
and  let  nA   =  A .     Then 


98  RESERVOIR  SLUICES. 


cc  being  the  coefficient  of  contraction. 

Thus  the  swell  will  be  found  to  be  further  from  or  nearer 
the  sluice,  according  as  the  difference  between  the  depths  of 
the  stream  and  the  sluice  is  >  or  <  BC. 

If/2  is  the  coefficient  of  hydraulic  resistance,  then 


and/2    may  be  .  I  or  even  greater;  but  if  the  sluice  edges  are 
smoothed  and  rounded  so  that/2  can  be  disregarded,  then 


and  therefore  AB  -  BC  =  AC  =  ^-. 

*g 
It  is  assumed  that  the  water  in  the  reservoir  retains  the 

same  level,  but  where  the  flow  commences  there  is  a  depres- 
sion in  the  surface  due  to  the  velocity  of  flow,  and  the  amount 
of  this  depression  should  be  deducted  from  the  total  head. 
When  the  backwater  rises  above  the  sluice,  as  in  Fig.  68, 

AC  =  head  required  to  produce  v2  -\-  head  "  lost  in  shock  " 


f   2 
^2 


and  AC  increases  with  «,  i.e.,  as  Al  diminishes  as  compared 
with  An. 


FLOW  OVER   WEIRS  (BAZIN}.  99 

27.  Bazin's  Flow  Over  Weirs.  —  This  article  is  the  resume  of  Bazin's  val- 
uable papers  on  this  subject  published  in  the  Annales  des  Fonts  et 
Chaussees.  The  symbols  are  changed  to  correspond  with  the  preceding; 
articles  of  the  present  chapter. 

Let  cs  ,  £s  ,  Hs  be  the  coefficient,  length  of  crest,  and  head  over  crest 
for  a  standard  weir. 

Let  c,  B,  H*  be  the  corresponding  symbols  for  an  experimental  weir, 

Then,  disregarding  the  velocity  of  approach, 


and 


Experiments  with  the  standard  weir  give  the  value  of  cs ,  the  ratio 

n  J-J 

— s  is  usually  unity,  and  the  ratio  — -  is  found  by    observation.     Hence 
Jt>  J~ii 

the  value  of  c  can  be  at  once  calculated. 

In  practice  it  seems  impossible,  with  the  data  at  present  available,  to 
make  a  rational  selection  of  the  proper  value  of  c,  which  varies  betweert 
wide  limits  and  is  affected  not  only  by  the  form  of  the  weir  but  by 
other  conditions,  amongst  which  may  be  enumerated  the  following  : — 

(a)  The  velocity  of  approach,  which  cannot  be  disregarded  when  the 

weir  is  of  small  height. 
(ft)  The  height  of  the  weir. 
(<?)  The  crest  contraction,  which  depends  both  upon  the  height  of  thes 

M'eir  and  the  form  of  the  crest. 

(d)  The  end  contractions,  which  have  a  considerable  influence  where 
•       the  weirs  are  of  comparatively  small   width,  but  are  not  of 

so  much  importance  when  the  weirs  are  long. 
(<?)  They^rw  of  the  nappe,  which  may  vary  considerably,  and  whiclu 

in  every  case  should  be  the  subject  of  a  careful  investigation. 

Sharp-crested  Weir  (Figs.  54,  55). — The  simplest  and  best  defined 
case,  and  one  which  admits  of  an  exact  determination  of  the  coefficient 
of  discharge  c,  is  that  of  a  free  nappe  (or  sheet),  the  sheet  of  water  flow- 
ing over  the  weir  without  end  contraction,  and  with  its  lower  as  well 
as  upper  surface  fully  exposed  to  atmospheric  pressure.  Allowance 
may  be  made  for  the  influence  upon  the  discharge  of  the  velocity  of 
approach,  u,  by  substituting  for  the  head,  H* ,  over  the  crest  in  the 

discharge  formula,  the  expression  H?  +   «  — ,  a  being  a  coefficient  whicha 

& 


100  FLOW  OVER   WEIRS  (B4ZIN). 

has  not  been  accurately  determined.     Thus 


LI  being  the  modified  value  of  c. 

But  — j  is  very  small,  rarely  exceeding  a  few  centimeters,  and  there- 
fore, approximately, 


Let  x  be  the  height  of  the  weir.     Then 

uB(H*  +  x)  =  O  =  cB 
and  therefore 

^77,  - c*  (T/TT^  - 

Hence,  putting    K  =  f «r2, 


so  that 


Bazin  has  deduced  the  values  of  a,  A",  and  //  by  comparative  experi- 
ments on  five  weirs  of  different  heights. 

a.  and  K  are  not  constant,  but  their  mean  values  are  f  and  .55 
respectively.  The  coefficient  //  slowly  diminishes  as  the  head  /;  in- 
creases. 

Thus 

for  heads  =  oni.o5     om.io    om.2o    otn.3o    ora.4o    om.5o 
the  correspond  ing  values  of  )J.  =  .448       .432       .421       .417       .414      .412 

For  values  of  H a  >  o^.io,  it  is  sufficiently  accurate  to  take 

.oo-i 
H  =  .405  +  -, 


FLOW  OYER   WEIRS  (BAZ1N}.  101 

and  therefore 


so  that 


Generally,  for  values  of  //2  between  om.io  and  om.3o,  jj.  may  be  made- 
equal  to  .425,  and,  taking  K  =  .5, 


a  suitable  form  for  practical  use  when  errors  of  2  to  3  per  cent  are  not 
too  large  to  be  of  importance. 

The  absolute  values  of  c  having  been  found  for  a  sharp-crested  weir 
with  a  free  nappe  and  a  vertical  face  on  the  up-stream  side,  it  does  not 
follow  that  the  same  method  should  be  adopted  to  determine  the  cor- 
responding coefficients  for  other  forms  of  weir.  In  fact,  if  c'  is  the  coef- 
ficient for  any  other  given  weir,  when  the  head  over  the  crest  is  the 
same,  the  influence  of  the  velocity  of  approach  may  be  largely  eliminated 

by  finding  the  ratio  —  .  The  ratio  corresponding  to  two  different  in- 
clinations is  sensibly  constant  for  all  heads,  and  the  following  table 
gives  the  values  of  -  for  varying  face-slopes:  — 

For  an  up-stream  face-slope  of  i  hor.  to  I  vert  .......    —  =    .93 

2         "      3     "    ......     "  =  .94 

i          "      3     "    .......   "  =  -96 

"  "  vertical  face  ......................   "  =  .00 

For  a  down-stream  face-slope  of  I  hor.  to  3  vert  .....    "  =  .04 

2         "     3    "...."=  .07 

1  I       "        ____     "    =      .10 

2  I      "       ____     "    =     .12 

4        "      I    "...."=    .09 

It  may  be  noted  that  the  coefficient  (or  ratio)  gradually  increases 
from  .93,  corresponding  to  a  slope  of  45°  on  the  up-stream  side,  to  1.12, 
corresponding  to  a  slope  of  about  30°  on  the  down-stream  side. 


302 


FLOW  OVER    WEIRS  (BAZIN). 


When  the  air  cannot  pass  uuderneath  the  sheet  of  water  flowing  over 
the  crest,  the  nappe  either  encloses  a  volume  of  air  at  less  than  the 
atmospheric  pressure  and  is  depressed,  Fig.  69,  or  the  air  is  entirely  ex- 


FIG.  69.  —  Depressed  Nappe.  FIG.  70.  —  Drowned  Nappe. 

eluded  and  the  nappe  is  wetted  underneath  or  drowned,  Fig.  70.  The 
latter  condition,  when  the  nappe  encloses  an  eddying  mass  of  fluid,  gives 
a  more  uniform  motion,  as  the  pressure  of  an  enclosed  volume  of  air 
may  vary  from  the  accidental  admission  of  new  air.  The  discharge  is 
slightly  greater  than  with  the  free  nappe,  and  may  be  increased  almost 
10  per  cent  when  the  nappe  is  on  the  point  of  being  drowned.  So  long 
.as  the  head  exceeds  a  certain  limit,  the  nappe  will  not  be  in  contact  with 
the  weir  face.  The  drowned  nappe  may  be  either  independent  of  or 
influenced  by  the  down-stream  level  according  as  the  rise  produced 
beyond  the.  nappe  is  at  a  distance  from  the  foot  of  the  nappe  or  partially 
-encloses  the  foot. 

Rise  at  a  Distance  from  the  Foot  of  the  Nappe.  —  In  this  case 

-  =  .878  +  .128  £, 

c  ^     Hi 

t>ut  the  max.  value  of  —  cannot  exceed  2|,  as  the  drowned  condition  no 

//a 

longer  holds  when  //3  <  "  x.    The  value  of  --.  corresponding  to  this  maxi- 

mum, is  1.2,  and  if  /7a  =  x,  the  coefficients  c'  and  rare  sensibly  the  same. 
Applying  this  formula  to  weirs  of  different  heights,  it  is  found  that  the 
absolute  values  of  the  coefficients  of  discharge  are  sensibly  given  by 
the  formula 

* 
c'  =  .47 


Rise  Enclosing  the  Foot  of  the  Nappe.  —  if  D  is  the  difference  of  level 
^between  the  weir-crest  and  the  down-stream  surface, 


FLOW  OfER    WEIRS  (BAZIN). 


103 


for   which  it  is  usually  sufficiently  accurate  to  substitute  the   simpler 
•expression 

c'  D 


These  formulae  are  only  true  for  values  of  D  between  certain  limits. 
If  Hi  +  D  is  greater  than  about  \x,  the  rise  is  moved  beyond  the  foot  of 
the  nappe,  and  the  formulae  in  the  preceding  case  become  applicable. 
Again,  if  the  head  Hi  is  not  sufficient  to  enable  the  nappe  to  push  back 
the  rise,  the  down-stream  surface  level  must  be  sufficiently  high  to  pre- 
vent the  admission  of  air  below  the  nappe. 

The  drowned  nappe  preserves  its  characteristic  profile  even  when  the 
down-stream  surface  is  on  a  level  with  the  weir  crest,  Fig.  71,  but  if 
the  difference  of  level  between  the  up-  and  down-stream  surfaces  still 
continues  to  diminish,  a  point  is  reached  at  which  the  nappe  suddenly 
and  with  an  undulating  movement  again  forms  part  of  the  surface.  This 
change,  which  is  very  apparent,  does  not  seem  to  have  much  influence 
on  the  coefficient  of  discharge. 


FIG.  71.— Drowned  Nappe. 


FIG.  72. — Adhering  Nappe  (fft 
not  very  small). 


FIG.  73. — Adhering  Nappe,  spring- 
ing clear  above  crest. 


FIG.  74. — Adhering  Nappe 
small). 


On  certain  rare  occasions,  and  under  conditions  governed  by  the 
thickness  of  the  weir  and  by  the  construction  of  the  upper  portion  carry- 
ing the  crest,  the  nappe  becomes  adherent,  Figs.  72  to  76.  the  sheet  of 
water  remaining  in  contact  with  the  weir  face.  The  coefficient  c'  is  then 


104 


FLOW  OVER   WEIRS  (BAZ1N). 


increased  and  may  become  as  large  as  1.3^,  corresponding  to  an  absolute- 
value  of  .55  or  .56. 

From  what  has  been  said  it  may  be  at  once  inferred  that  the  dis- 
charge over  a  weir  is  largely   influenced   by  the   form   of  the   nappe_ 


FIG.  75. — Nappe  adhering  on  crest  only. 


FIG.  76. 


Taking,  for  example,  a  sharp-crested  weir  0.75  m.  high,  it  was  found 
that  for  a  head  over  the  crest  of  0.2  m.  the  coefficient  of  discharge  c 
was 

.433  for  a  free  nappe. 

.46     "    a  depressed  nappe. 

.497    "    a  drowned  nappe. 

.554   "    an  adhering  nappe. 

Beam  Weirs. — These  weirs  are  formed  of  squared  timbers  laid  one 
above  the  other  to  any  required  height,  the  weir  faces  being  vertical  and 
the  crest,  or  sill,  having  a  width  e  equal  to  that  of  the  timbers. 

Free  Nappe. — The  nappe  may  either  spring  clear  from  the  up-stream 
edge,  when  the  case  becomes  that  of  a  sharp-crested  weir,  or  it  may 
remain  in  contact  with  the  sill  and  spring  clear  from  the  down-stream 
edge.  The  first  case  is  at  once  realized  4f  H*  exceeds  2<?,  and  may  occur 
for  any  value  of  h  between  2<?  and  \\e,  the  change  being  produced  by 
any  such  extraneous  disturbing  cause  as  the  admission  of  air  or  the 
passage  of  a  floating  body,  etc. 

When  the  nappe  remains  in  contact  with  the  sill, 


TT 

an  expression  depending  essentially  on  the  value  of  —  . 


•For?- 


.5.  ...=.  79 


=1.0 
=1.5 
=2.0 


=    .88 

=    .98  )  If  the  nappe  remains  in 

=  1.07  \      contact  with  the  sill. 


FLOW  OVER   WEIRS  (BAZIN).  105 

J  TT 

The  ratio  -  is  unity  for  all  values  of — -  above  2,  and  if  the  nappe 

»  c  £ 

H 

springs  clear  from  the  up-stream  edge,  for  all  values  of — 2  between  \\ 

and  2. 

With  sills  of  considerable  width,  e.g.,  I  or  2  m.,  the  above  formula 

TT 

still  gives  results  which  are  approximately  correct.  The  ratio — 2  may 
diminish  to  a  few  tenths  or  even  less  than  .35.  With  a  2-m.  flat-crested 
weir  experiment  gave  for  a  head  of  .45  m.,  —  =  .755,  the  corresponding 

absolute  value  of  c'  being  .337.  The  formula  gives  —  =  .742,  the  corre- 
sponding value  of  c'  being  .331. 

The  rounding  of  the  up-stream  edge  of  the  sill  has  a  very  sensible 
influence  upon  the  flow,  and  the  effect  of  a  radius  of  only  i  or  2  cm.,  as 
usually  results  from  ordinary  wear,  must  by  no  means  be  disregarded 
in  gauging  the  discharge.  Fteley  and  Stearns  observed  that  the  effect 
of  a  small  radius  R,  not  exceeding  £  in.,  or  0.012  m.,  was  to  increase 
the  head  by  .7 A',  and  therefore  the  coefficient  c'  in  the  ratio  of 

H-?  to  (//a  +  -7 A5)',  or  approximately  I  to  I  +  -—  .       This  approxima- 

-T/2 

tion  is  not  sufficiently  accurately  for  sensibly  greater  radii.  With  two 
weirs,  the  one  .8  m.  and  the  other  2  m.  wide,  the  up-stream  edges  being 
rounded  to  a  radius  of  .10  m.,  the  discharge  was  increased  14  per  cent  in 
the  first  and  12  per  cent  in  the  second  case.  With  the  2-m.  weir  the 
coefficient  c'  for  the  greatest  head  used  in  the  experiments  was  found  to 
be  .373,  which  is  very  nearly  the  same  as  the  value  theoretically  deduced 
on  the  assumption  that  the  flow  over  the  weir  is  in  fluid  filaments  par- 
allel to  the  sill.  This  condition  is  only  imperfectly  realized  in  practice 
as  the  surface  of  the  nappe  invariably  has  an  undulatory  movement. 

Depressed  and  Drowned  Nappes. — With  a  sharp-crested  weir  the  co- 
efficient for  a  depressed  nappe  is  always  greater  than  that  for  a  free 
nappe.  With  a  beam  weir,  such  as  that  now  under  consideration,  the 
coefficients  differ  only  slightly,  that  for  the  depressed  weir  being  at  first 
a  little  less,  then  about  the  same,  and  finally  a  little  greater  than  the 
coefficient  for  the  free  nappe.  When  the  nappe  is  drowned,  the  influence 
of  contact  with  the  sill  is  complicated  by  the  fact  that  it  is  impossible  to 
define  exactly  the  point  at  which  the  nappe  is  freed  from  the  sill,  and 

TT 

this  separation  no  longer  corresponds  to  a  certain  constant  value  of  — 2. 

It  may  again  occur  either  before  or  after  the  establishment  of  the  drowned* 
condition.  Two  cases  may  be  distinguished.  If  x  (the  height  of  weir) 
>  5<?,  the  separation  takes  place  in  advance  of  the  drowned  state,  and  in 
this  intermediate  condition  the  nappe  does  not  differ  from  that  which. 


lo6  FLOW  OVER   WEIRS  (B4ZIN). 

flows  over  a  sharp-crested  weir.  If  x  <  5<?,  the  nappe  is  not  freed  from 
the  sill  before  it  assumes  the  drowned  form,  and  at  the  moment  of  the 
change  is  very  unstable. 

So  long  as  the  contact  with  the  sill  continues,  its  influence  predomi- 
nates, and  the  formula 


is  fairly  applicable  to  the  drowned  nappe. 

But  when  the  nappe  has  left  the  sill,  the  phenomenon  becomes  more 
and  more  nearly  the  same  as  for  a  sharp-crested  weir,  and  the  formula 
now  applicable  is 


These  two  formulae  give  the  same  value  for  c'  for  a  certain  limiting 
value  of  Hi,  given  by 


The  first  formula  holds  when  the  heads  are  less  than  H*  ',  but  the  co- 
efficients are  a  little  too  small  although  the  errors  are  never  more  than 
3  or  4  per  cent.  If  the  heads  are  greater  than  HJ,  the  second  formula 
is  to  be  used,  but  the  results  are  again  too  small  and  the  error  in  this 
case  may  be  as  much  as  8  per  cent  at  the  moment  when  the  nappe  is 
separated  from  the  sill.  The  error  then  rapidly  diminishes  as  the  head 
increases. 

Wide-crested  Weirs  with  Sloping  Faces.  —  In  these  the  coefficient  </, 
depending  upon  the  head  (Hi),  the  width  (<?)  of  crest,  and  the  degree  of 
face-slope,  is  now  extremely  variable  and  each  case  must  be  sub- 
jected to  a  special  investigation.  The  face-slope  on  the  up-stream  side 
has  the  effect  of  diminishing  the  contraction  and  therefore  increasing 
the  discharge.  The  down-stream  face-slope,  on  the  other  hand,  pro- 
duces an  effect  similar  to  the  widening  of  the  crest  and  diminishes  the 
discharge.  The  rounding  of  the  up-stream  edge  of  the  crest  consider- 
ably diminishes  the  contraction  and  may  increase  c'  by  10  or  15  per  cent. 
Finally,  c'  is  very  largely  increased  for  weirs  with  completely  curved  pro- 
files. 

Bazin  has  prepared  Tables  comprising  a  sufficient  number  of  partic- 
ular cases  which  may  serve  as  a  guide  in  practice.  It  is  impracticable 
to  establish  a  general  formula  which  will  take  into  account  all  the  vari- 
able elements  referred  to. 

Drowned  Weirs  with  Sharp  Crests.  —  When  the  water  on  the  down- 
stream side  does  not  stand  much  above  the  crest  of  the  weir,  Bazin  gives 
the  somewhat  complicated  formula 

•     c1  \D       (  i  D       i  (DV  }    x 

-  -  1.  06  +   ---  -J  .008  +  --  +  -—       \-jT- 

c  4*1  3  x      3  \xj    J  //2 


FLOW  OVER   WEIRS  (BAZ1N}.  107 

In  the  majority  of  cases,  however,  the  following  simpler  formula  is 
-applicable : 


These  two  formulae,  established  so  as  to  represent  as  accurately  as 
possible  the  particular  experiments  by  which  they  have  been  deduced, 
may  be  replaced  by 

C'  I  // 

=      ,.05    +    .3,     - 


which  will  give  results  differing  from  those  obtained  with  the  other  for- 

TT  TT    I 

mulae  by  not  more  than  about  i  or  2  per  cent,  unless  —  and  -  —  are 

very  small,  when  the  difference  may  be  as  much  as  4  or  5  per  cent,  but 
in  the  latter  case  the  determination  of  c'  is  always  somewhat  uncertain. 
The  effect  of  drowning  is  not  the  same  for  wide-crested  weirs.  The 
flow  on  the  up-stream  side  is  not  affected  by  the  depth  of  the  water  on 
the  down-stream  side  until  the  down-stream  surface  rises  considerably 
above  the  weir  crest,  and  the  effect  diminishes  as  the  width  of  the  crest 
increases.  In  the  case  of  a  sharp-crested  weir  the  influence  upon  the 
up-stream  flow  is  felt  before  the  down-stream  surface  has  reached  the 
level  of  the  crest.  As  the  width  of  a  wide-crested  weir  increases  it  loses 
its  weir  characteristics  and  approximates  more  and  more  closely  to  an 
^>pen  channel  with  horizontal  bed. 

Thickness  of  Nappe  on  Weir  Crest.  —  Let  /  =  thickness  of  nappe. 

For  a  sharp-crested  weir  and  free  nappe  -—  -  varies  from  .85  to  .86. 

//2 

For   a   sharp-crested  weir   and    drowned  nappe  77  increases  with  -» 
being  .8  when  •-  =  .4,  .855  when  -  =  i,  and  .87  when    —  >  i.     As  the 

down-stream  level  rises  -,-    increases,  exceeding  .9   for   the    undulating 

condition,  and    necessarily  tends    to    unity  as    the    difference  of   level 
between  the  crest  and  the  down-stream  surface  is  greatly  diminished. 

In  beam  weirs  with  free  nappes  -  >  .9  for  small  values  of  —  ,  and  de- 

cieases  as  the  head   increases  until   the  ratio  becomes  .855,  when  the 
nappe  is  on  the  point  of  separating  from  the  sill. 

In  beam  weirs  with  drowned  nappes  the  variation  of  —    is  somewhat 

//2 

complicated.     The  ratio  diminishes  until  a  minimum   is  reached,  and 


io8 


BERNOULLI'S    THEOREM. 


then  increases  and  approximates  to  values  which  are  the  same  as  in  the* 
case  of  sharp-crested  weirs. 

In  wide-crested  weirs  with  sloping  faces  —  is  very  variable.  Gener- 
ally it  increases  as  the  down-stream  slope  diminishes,  and  diminishes 
with  the  up-stream  slope.  In  weirs  in  which  the  crest  is  connected  with 

the  up-stream  face  by  a  curved  surface  — —  may  be  less  than  .8,  but  the 

I  ,     determination  of  the  nappe  thickness  is  in 

such  case  much  less  accurate. 

28.  Bernouilli's  Theorem. — A  simple  proof 
of  this  theorem  is  as  follows: 

Consider  an  indefinitely  small  element  of 
a  stream-liife,  of  length  ds  and  sectional 
area  a. 

Let  p,  p  +  dp  be  the  intensities  of  pressure 

at  the  ends. 

"    TV  be  the  specific  weight  of  the  fluid. 
"    a.    "     "    angle  between  the  direction 
of  motion  of  the  element  and  the 
vertical. 
"    dz  be  the  vertical    projection  of   ds^ 

so  that  dz  =  ds .  cos  a. 
Resolving  in  the  direction  of  motion, 

ds  cos  a  =  accelerating  force 

=  mass  x  acceleration 

w         .      dv 
=  —  a  .  ds  .  — 

g  dt 

^     ii)  dv 

=  —a  .  v  .  dt  .  — 


pa  —  (p  x  dp)a  —  wa 


IV 

=  —av  .  dv. 


»                **' 
.  *.  —  a  .  dp  —  iva  .  dz  = av  .  dv, 

£ 

dp      v  .  dv 

or  dz  -\ +  -     —  =  o. 

w  g 


Integrating,  z  +     I  -  -  +     -    =  a  const.,   is  true  for   any  fluid.     If 
the  fluid  is  water,  w  is  constant,  and  then  z  +  2- •  +  —  —  a  const. 


EXAMPLES.  109 


EXAMPLES. 

(N.  B.   In  the  following  examples  #-  =  32  unless  otherwise  specified.) 

1.  7"  tons  of  water  fall  H  feet  per  minute  and  are  employed  to  turn 
turbines  which  transform  into  useful  work  orte  half  of  the  total  energy 

x>f  the  water.     What  is  the  H.P.  of  the  turbines  ?  TH  ^ 

AHS.  .    ' 

33 

2.  A  turbine  transforms  into  9.72    H.P.    of  useful  work  the  energy 
•of  the  water  fall  ing  jf  feet  from  a  Thomson  V-notch  in  which  the  water 
stands  at  a  constant  level  fcjtft.  above  the  bottom  of  the  notch.     If  the 
-coefficient  of  discharge  is  .o,*what  is  the  efficiency  of  the  turbine? 

Ans.  .8. 

3.  A  fall  of  10  ft.  supplies  to  a  turbine  12  cu.  ft.  of  water  per  sec. 
The  turbine  uses  only  8  ft.  of  the  fall,  and  the  water  leaves  the  turbine 
with  a  velocity  of  8  ft.   per  sec.     If  500  Ibs.-ft.  are  lost  in  frictional  re- 
sistance, etc.,  find  the  efficiency  of  the  turbine.  Ans.  .634.^ 

4.  10,000  5o-volt  incandescent  and   250  45o-watt  arc  lamps  are  to  be 
supplied  with  power  from  a  waterfall  having  an  effective  head  of  40  ft., 
~2Q  miles  distant.     Losses  between  lamps  and  converting  apparatus  at 
receiving  end  of  transmission,  5$;  ^efficiency  of  converting  apparatus, 
•92$  ^line    losses,    10$  ;;  losses  in   generators  and    transformers  between 
line  and  turbine  shaft,  io^-f  efficiency  of  turbine,  85^.     Required,  neces- 
sary flow  of  water  per  hour.  Ans.  1,080,630  cu.  ft.  per  hour. 

5.  A  frictionless  pipe  gradually  contracts  from  a  6-in.  diameter  at  A 
to  a  3-in.  diameter  at  B,  the  rise  from  A  to  B  being  2  ft.     If  the  delivery 
is  i   cu.  ft.  per  second,  find  the  difference  of  pressure  between  the  two 
points  A  and  B.  Ans.  504.6  Ibs.  per  sq.  ft. 

6.  In  a  frictionless  horizontal  pipe  discharging  10  cu.  ft.  of  water  per 
second,  the  diameter  gradually  changes  from  4  in.  at  a  point  A  to  6  in. 
at  a  point  B.     The  pressure  at  the  point  B  is  100  Ibs.  per  square  inch  ; 
find  the  pressure  at  the  point  A.  Ans.  4118  Ibs.  per  sq.  ft. 

7.  A  ^-in.  horizontal  pipe  is  gradually  reduced  in  diameter  to  £  in. 
-and  then  gradually  expanded  again  to   its  mouth,  where   it  is  open  to 
the   atmosphere.     Determine   the   maximum    quantity  of  water  which 
can  be  forced  through  the  pipe  (a)   when  the  diameter  of  the  mouth 
is  ^  in.,  (b)  when  the  diameter  is  f  in.    Also  determine  the  corresponding 
velocities  at  the   throat  and  the  total   heads   (neglect   friction,  which, 
however,  is  very  considerable). 

Ans.  (a)  .24  cu.  ft.  per  min.  ;  46.7  ft.  per.  sec. 
(b)  .239  cu.  ft.  per  min. ;  46.66  ft.  per  sec. 


no  EXAMPLES. 

8.  A  short  horizontal  pipe  ABC  connecting  two  reservoirs  gradually 
contracts  in  diameter  from  i  in.  at  A  to  £  in.  at  j#and  then  enlarges  to. 
i  in.  again  at  C.     If  the  height  of  the  water  in  the  reservoir  over  C  be 
12  ins.,  determine  the  maximum  flow  through  the  pipe  and  sketch  the 
curve  of  pressures.     Also  obtain  an  equation  for  this  curve,  assuming 
the  rates  of  contraction  and  expansion  of  the   pipe   to  be  equal  and 
uniform.  Ans.  4  cu.  ft.  per  min. 

9.  In  a  diverging  mouthpiece  the  diameter  of  the   throat  is  .6  in.,, 
and  the  head  of  water  over  the  axis  is  30  ft.     What  is  the  discharge  in. 
gallons  per  minute  when  the  vacuum  at  the  throat  is  18.3  ins.  of  mer- 
cury ?  Ans.  42. 

:"'  10.  In  a  stream  with  still  water  240  ft.  above  datum  and  flowing 
without  friction,  the  velocity  at  a  point  15  ft.  above  datum  is  24  ft.  per 
second.  What  is  the  pressure  at  this  point?  9 'j£  <F" 

Ans^.\Q$i$5  Ibs.  per  sq.  in. 

11.  A  funnel-shaped    mouthpiece    leads  from  a  reservoir  into  a  6-in. 
frictionless  pipe,  so  that  there  is  no  contraction.     The  water  flows  with 
a  velocity  of  24  ft.   per  second.     Find  the  pressure  at   a   point  in  the: 
pipe  10  ft.  below  the  surface  of  the  water  in  the  reservoir. 

Ans.  15.43  Ibs.  persq.  in. 

12.  A  3-in.  pipe  gradually  expands  to  a  bell-mouth  ;  if  the  total  head, 
//,  be  40  ft.,  find  the  greatest  diameter  of  the  mouth  at  which  it  will 
run  full  when  open  to  the  atmosphere.     Compare  the  discharge  from 
this  pipe  with  the  discharge  when  the  pipe  is  not  expanded  at  the  mouth, 

Ans.  4.8   in. ;  discharge  is  149.076  cu.  ft.  per  minute  with  bell- 
mouth  and  47.345  cu.  ft.  per  minute  without  bell-mouth. 

13.  The  pressure  in  a  12-in.  pipe  at  A  is  50  Ibs.  per.  sq.  in.  ;  the  pipe 
then  enlarges  to  a  15-in.   pipe  at  B,  the  rise  from  A  to  B  being  3  ft.  i 
the  discharge  is  noo  cu.  ft.  per  minute.     Find  the  pressure  at  B  \  also 
find  the  pressure  at  a  point  C,  the  rise  from  B  to  C  being  6  ft. 

Ans.  7142^  Ibs/per  sq.  ft.;  6767^  Ibs.  per  sq.  ft. 

14.  One  cubic  foot  of  water  per  second   flows   steadily  through  a, 
frictionless  pipe.     At  a  point  A,  100  ft.  above  datum,  the  sectional  area 
of  the  pipe  is  .125  sq.  ft.,  and  the  pressure  is  2500  Ibs.  per  sq.  ft.     Find 
the  total  energy.     At  a  point  B  in  the  datum-line  the  pressure  is  1250 
Ibs.  per  sq.  ft.  and  the  sectional  area  is  .0625  sq.  ft.     Find  the  loss  of 
energy  between  A  and  B.     Find  the  "  loss  in  shock,"  if   the  sectional 
area  at  B  abruptly  changes  (a)  from  .125  to  .0625  sq.  ft.  ;  (b*)  from  .0625 
to  .125  sq;  ft. 

Ans.  141  ft. -Ibs.  ;  117  ft.-lbs.  ;  79  ft.-lbs.  per  cu.  ft.  ;  62^  ft.-lbs. 
per  cu.  ft. 

15.  In  a  frictionless  pipe  the  diameter  gradually  changes  from  6  inv 
at  a  point  A  20  ft.  above  datum  to  3  in.  at  B  15  ft.  above  datum.     The 
pressure  at  A  is  20  Ibs.  per  sq.  in. ;  find  the  pressure  at  B,  the  delivery 
of  the  pipe  being  2f  cu.  ft.  per  sec.  Ans.  2.23  Ibs.  per  sq.  in. 


EXAMPLES.  1 1 1 

16.  A   horizontal   frictionless   pipe  gradually  contracts  to  a  throat 
of  *ith  of  the  area  and  then  gradually  enlarges  again  to  a  pipe  of  the 

same  size.     If  V  is  the  velocity  of  flow  in  the  pipe,  find  the  reduction  of 

pressure  at  the  throat.  WVZ 

Ans.   •(«'  -  i). 

17.  The  pressure  in  a  3^-in.  horizontal  frictionless  pipe  is  62|  Ibs.  per 
sq.  in.  above  that  of  the  atmosphere.     The  pipe  is  gradually  reduced  to- 
a  throat  of  one  fifth  of  the  area  and  discharges  into  the  atmosphere. 
Find  the  velocity  of  efflux  and  the  amount  of  the  discharge  in  gallons 
per  minute.  Ans.  97.98  ft. 'per  sec.  ;  491.177  gals. 

1 8.  A  frictionless  play-pipe  gradually  expands  from  a  diam.  of  I  in.  at 
the  base  to  a  diam.  of  3  in.  at  the  mouth.     There  is  a  discharge  of  33  cu. 
ft.  per  min.  under  a  head  of  183  feet.     Find  the  coefficient  of  discharge, 
the  force  required  to  hold  the  nozzle,  and  the  total  H.P.  developed. 

Ans.  .9265;   108.11  Ibs.;   11.56  H.P. 

19.  Find  the  discharge  in  cubic  feet  per  minute  under  a  head  of  2  it. 
through  a  horizontal  frictionless  pipe  which  gradually  diminishes  from 
a  diam.  of  \  in.  to  a  throat  of  \  in.  diam.,  at  which  the  pr.  head  =  6  ins., 
and  then  gradually  enlarges  to  a  pipe  of  same  diameter  as  before. 

Ans.  .2017. 

•  20.  Find  the  head  required  to  give  i  cu.  ft.  of  water  per  second 
through  an  orifice  of  2  square  inches  area,  the  coefficient  of  discharge 
being  .625.  (^=32.)  Ans.  207.36  ft. 

21.  The  area  of  an  orifice  in  a  thin  plate  was  36.3  square  centimetres, 
the  discharge  under  a  head  of  3.396  metres  was  found  to  be  .01825  cubic 
metre  per  second,  and  the  velocity  of  flow  at  the  contracted  section,  as 
determined  by  measurements  of  the  axis  of  the  jet,  was  7.98  metres  per 
second.     Find  the  coefficients  of  velocity,  contractien,  discharge,  and 
resistance.     Gr=9.8i.)  Ans.  .977;   .632;  .616;  .046. 

22.  The   piston  of  a  12-in.  cylinder  containing  salt-water  is  pressed 
down  under  a  force  of  3000  Ibs.     Find  the  velocity  of  efflux  and  the 
volume  of  discharge  at  the  end  of  the  cylinder  through  a  well-rounded 
i-in.  orifice.     Also  find  the  power  exerted,  cv  being  .977  and  c  =  .5343. 

Ans.  60.373  ft.  per  sec. ;  .  i76'cu.  ft.  per  sec. ;  1.166  H.P. 

23.  In  the  condenser  of  a  marine  engine  there  is  a  back  pressure  of 
26|  in.  of  mercury;  the  injection  orifices  are  6  ft.  below  the  sea-level. 
With  what  velocity  will  the  injection-water  enter  the  condenser  ?     (Neg- 
lect resistance  and  take,?-—  32.2.)  Ans.  25.3  ft.  per  sec. 

24.  Water  in  the  feed-pipe  of  a  steam-engine  stands  12  ft.  above  the 
surface  of  the  water  in  the  boiler  ;  the  pressure  per  sq.  in.  of  the  steam  is 
20  Ibs.,  of  the  atmosphere  15  Ibs.     Find  the  velocity  with  which  the 
waler  enters  the  boiler,  cv  being  .97.  Ans.  5.376  ft.  per  sec. 

2-5.  The  injection  orifice  of  a  jet  condenser  is  5  ft.  below  sea-level 


H2  EXAMPLES. 


vacuum  =  27  in.  of  mercury.     Find  velocity  of  water  entering  con- 
denser, supposing  three  fourths  of  the  head  lost  by  frictional  resistance. 

Ans.  23.86  ft.  per  sec. 

26.  The  jet  from  an  orifice  of  .008  sq.  ft.  area  in  the  side  of  a  tank 
-and  under  a  head  of  16  ft.  issues  horizontally  and  falls  1  ft.  vertically  in 
a  horizontal  range  of  7.68  feet.     The  delivery  is  60  gallons  per  minute. 
Find  the  coefficients  of  velocity,  discharge,  contraction,  and  resistance. 

Ans.  .96;  .625  ;  .65  ;  .085. 

27.  The  jet  from  a  circular  sharp-edge  orifice  £  in.  in  diani.  under  a 
liead  of    18  ft.,  strikes  a  point  at  a   distance  from   the  orifice  of  5  ft. 
measured     horizontally    and    4.665    ft.    measured    vertically.     The    dis- 
charge is  98.987  gallons  in   569.218  seconds.     Find  the  coefficients   of 
discharge,  velocity,  contraction,  and  resistance. 

Ans.  .6009;  .945;  .635;  .1196. 

28.  A  sluice  3  ft.  square  and  with  a  head  of  12  ft.  over  the  centre  has, 
from  the  thickness  of  the  frame,  the  contraction  suppressed  on  all  sides 
when  fully  open  ;    when  partially  open,  the  contraction    exists  on   the 
upper  edge,  i.e.,  against  the  bottom  of  the  gate,  which  is  formed  of  a 
thin  sheet  of  metal.     Find  the  discharge  in  cubic  feet  when  opened  i  ft.t 
2  ft.,  and  also  when  fully  open.  Ans.  57.22;  113.38;  173.51. 

29.  A  vessel  containing  water  is  placed  on  scales  and  weighed.     How 
will  the  weight  be  affected  by  opening  a  small  orifice  in  the  bottom  of 
the  vessel  ? 

30.  Water  is  supplied  by  a  scoop  to  a  locomotive  tender  at  7  feet 
above  trough.     Find  lowest  speed  of  train  at  which  the  operation   is 
possible.  Ans.   14.44  rniles  per  hour. 

Also  find  the  velocity  of  delivery  when  train  travels  at  40  miles  per 
hour,  assuming  half  the  head  lost  by  frictional  resistance.     (cv  =  i.) 

Ans.  35.68  ft.  per  second. 

31.  The  head  in  a  prismatic  vessel  at  the  instant  of  opening  an  orifice 
was  6  ft.  and  at  closing  it  had  decreasecl  to  5  ft.     Determine  the  mean 
constant  head  h  at  which,  in  the  same  time,  the  orifice  would  discharge 
the  same  volume  of  water.  Ans.   5.488  ft. 

32.  A  cylindrical  vessel  5.747  in.  in  diameter  has  an  orifice  of  .2  in. 
diam.  at  the  bottom  ;  the  surface  sinks  from  16  in.  to  12  in.  in  53  seconds. 
Find  the  coefficient  of  discharge.  Ans.  .6. 

33.  A  prismatic  basin  with  a  horizontal  sectional  area  of  9  sq.  ft.  has 
an  orifice  of  .9  sq.  in.  at  the  bottom  ;  it  is  filled  to  a  depth  of  6  ft.  above 
the  centre  of  the  orifice.     Find  the  time  required  'for  the  surface  to  sink 
2  ft.,  3^  ft.,  5  ft.  Ans.  258.9  sec.  ;  500.16  sec.  ;  834.8  sec. 

34.  The  water  in  a  cylindrical  cistern  of  144  sq.  in.  sectional  area  is 
16  ft.  deep.     Upon   opening  an  orifice  of  r  sq.  in.  in    the  bottom  the 
water  fell  7  ft.  in  i  minute.     Find  the  coefficient  of  discharge.     The  co- 
efficient of  contraction  being  .625,  find  the  coefficients  of  velocity  and 
resistance.  Ans.  .6;  .96;  0.85. 


EXAMPLES.  113 

35.  How  long  will  it  take  to  fill  a  paraboloidal  vessel  up  to  the  level 
of  the  outside  surface  through  a  hole  in  the  bottom  2  feet  under  water? 
4g  =  32  and  c  =  .625.) 

1  76  J/~  ^ 

^«j-.  J     -^—  B  being  the  parameter  of  the  parabola  and  ,4  the 

i  O^         ./i 

sectional  area  of  the  orifice. 

36.  How  long  will  it  take  to  fill  a  spherical  vessel  of  radius  r  up  to  the 
level  of  the  outside  surface  through  a  hole  of  area  A  at  the  lowest  point 
-and  2  ft.  under  water,  c  being  .625  ? 

AnS'  >r  ~~  6<- 


37.  A  loo-gallon  tank  is  100  feet  above  ground  and  is  filled  by  a  i^- 
inch   pipe  connected   with  an  accumulator   having  a   3-ft.   cylr.    piston 
loaded  with  50  tons.      If  the  mean  lift  of  the  piston  is  10  ft.  and  if  ^  of 
the  head  is  lost  in  frictional  resistance,  how  long  will  it  t^.ke  to  fill  the 
tank?  Ans.  14.49  sees. 

38.  A  bucket  of  water  in  a  balance  discharges  4  Ibs.,  of  water  per 
minute  through  an  orifice  in  its  base  at  45°  to  the  vertical,  and  is  kept 
constantly  full  by  a  vertical  stream  which  issues  from  an   orifice   8  ft. 
above  the  surface  with  a  velocity  of  30  ft.    per  sec.     Show  that  the 
bucket  must  be  counterpoised  by  about  .066  Ib.  more  than  its  weight. 

39.  The  water  in  a  vessel  9  ft.  in  height  and  2  ft.  in  diameter  is  8  ft. 
deep.     In  what  time  would  one  half  of  the  water  flow  away  through  an 
orifice  in  the  bottom  i  inch  in  diameter?     If  the  orifice  is  closed  and 
the  vessel  is  made  to  rotate  about  its  axis  at.  the  rate  of  76T4T  revolutions 
per  minute,  to  what  height  will  the  water  rise  on  the  vessel's  surface? 
If  the  orifice  is  opened,  find  velocity  of  efflux  when  the  surface  at  the 
axis  is  3  ft.  above  the  orifice.     Also  find  the  difference  of  pressure-head 
in  a  horizontal  plane  6  inches  from  the  axis. 

Ans.  190.77  sees.;  to  the  top  ;   16  ft.  per.  sec.;  3  ins. 

(s  40.  A  cylindrical  vessel,  10  ft.  high  and  i  ft.  in  diameter,  is  half  full 
of  water.  Find  the  number  of  revolutions  per  minute  which  the  vessel 
must  make  so  that  the  water  may  just  reach  the  top,  the  axis  of  revolu- 
tion being  (i)  coincident  with  the  axis  of  the  vessel,  (2)  a  generating 
line  of  the  vessel.  Ans.  (i)  483;  (2)  241!. 

41.  A  vessel  full  of  water  weighs  350  Ibs.  and  is  raised  vertically  by 
means  of  a  weight  of  450  Ibs.     Find  the  velocity  of  efflux  through  an 
orifice  in  the  bottom,  the  head  being  4  ft.  and^  =  32.2. 

Ans.  17.02  ft.  per  sec. 

42.  A  vessel  full   of    water  makes  100   revols.  per   min.     Find  the 
velocity  of  efflux  through  an  orifice  2  ft.  below  the  surface  of  the  water 
•at  the  centre,  the  diam.  of  the  vessel  being  3  ft.  and  cv  =  i. 

Ans.  33.4  ft.  per  sec. 
./    What  will  be  the  velocity  if  the  vessel  is  at  rest? 

Ans.  11.3  ft.  per  sec. 


114  EXAMPLES. 

43.  Show  that  when    the  water  flowing  over  has  a 
depth  greater  than  .3874  ft.  it  is  carried  completely  over 
the   longitudinal   opening,    .83   ft.    in    width.     At  what 
depth  does  all  the  water  flow  in  ?  Ans.  .221  ft. 

44.  A    square    box    2    ft.  in  length  and   2  ft.  across  a 
diagonal  is  placed  with  a  diagonal  vertical  and  filled  with  water.     How 
long  will  it  take  for  the  whole  of  the  water  to  flow  out  through  a  hole 
at  the  bottom  of  .02  sq.  ft.  area  ?  (c  =  .625.)  Ans.  97.52  sees. 

45.  A  pyramid  2  ft.  high,  on  a  square  base,  is  inverted  and  filled 
with  water.      Find  the  time  in  which  the  water  will  all   run  out  through 
a  hole  of  .02  sq.  ft.  at  the  apex.     A  side  of  the  base  is  i  ft.  in  length. 
(c  =  .625.)  Ans.  5.656  sec 

46.  Find  the  discharge  under  a  head  of  25  ft.  through  a  thin-lipped 
square  orifice  of  i  sq.  in.  sectional  area,  (a)  when  it  has  a  border  on  one 
side,  (b)  when  it  has  a  border  on  two  sides. 

Ans.  (a)  .3576  cu.  ft.  per  sec.;  (b)  .3706  cu.  ft.  per  sec. 

47.  A  vessel  in  the  form  of  a  paraboloid  of  revolution  has  a  depth  of 
16  in.  and  a  diam.  of  12  in.  at  the  top.     At  the  bottom  is  an  orifice  of 
i  sq.  in.  sectional  area.     If  water  flows  into  the  vessel  at  the  rate  of  2-^ 
cubic  feet  per  minute,  to  what  level  will  the  water  ultimately  rise?   How 
long  will  it  take  to  rise  (a)  1 1  in.,  (b)  1 1.9  in.,  (c)  1 1.99  in.,  (d)  12  in.  above 
the  orifice?     If  the  supply  is  now  stopped,  how  long  (e)  will  it  take  to 
empty  the  vessel  ? 

Ans.  12  inches;  (a)  49.17  sec.;  (b)   124.2  sec.;  (c)  202  sec.  ;  (a) 
an  infinite  length  of  time  ;  (e)  11.3  sec. 

48.  If  the  vessel  in  Example  47  is  a  sphere   i  ft.  in  diameter,  to  what 
height  will  the  water  rise  ?     How  long  will  it  take  for  the  water  to  rise  (a} 
n  in.,  (b)  12  in.  above  the  orifice?     Howjong  (c)  will  it  take  to  empty  the 
vessel  ?  Ans.  12  inches  ;  (a]  67.16  sec.;  (b)  81.46  sec.;  (c)  24.13  sec. 

49.  In  a  vortical  motion  two  circular  filaments  of  radii  r\  ,  r2 ,  of  ve- 
locities «/i ,  s/a ,  and  of  equal  weight  W  are  made  to  change  place.    Show 

v* 
that  a  stable  vortex  is  produced  if  —  =  const.;  and  if  r*  >  r\  ,  show  that 

the  surfaces  of  equal  pressure  are  cones.  fj 

50.  Sometimes    the  crest  of   a   dam  is  raised  by 
floating  a  stick  L  into  the  position  Li ,  where  it  is 
supported    against  the   verticals.     The   stick  then 
falls  of   itself    into    position  L9  and    rests   on    the 
crest.     Explain  the  reason  of  this. 

51.  A  6-in.  pipe    discharges   8000  gals,  per  hour 

into  a  9-in.  pipe.    Find  the  loss  of  head  at  the  June-  FIG.  78. 

tion.  Ans.  1.58  ft. 

52.  Prove  that  for  a  Borda's  mouthpiece  running  full  the  coefficient 

i 
of  discharge  is  — . 


EXAMPLES.  115 

53.  Find  the  discharge  in  pounds  per    minute   through    a    Borda's 
rnouthpiece    I    in.  in  diameter,  the  lip  being    12    in.  below  the  water- 
surface,  (a)  when  the  jet  springs  clear  from  the  edge,  (b)  when  the  mouth- 
piece runs  full.  Ans.  (a)  81.845  \  (^)  1 15-74- 

54.  The   surface  of  the  water  in  a  tank  is  kept  at  the  same  level  ; 
obtain  the  discharge  at  60  in.  below  the  surface  (a)  through  a  circular 
orifice  i  sq.   in.   in  area,  (b)  through  a  cylindrical  ajutage  of  the  same 
sectional  area  fitted  to  the  outside,  (c)  through  the  same  ajutage  fitted 
to  the  inside,  and  determine  the  mechanical  effect  of  the  efflux  in  each 
case. 

Ans.  (a]  4.85    Ibs.  per  sec.;  20.536  ft.-lbs.  per  sec. 

(b)  6.366     "       "       "    ;  21.404      " 

(c)  5.49       "       "       "    ;  13.725      "  "     "     if  running  full. 
3.69       "       "       "    ;  16.638      "          "     "     if  jet  springs  clear. 

55.  Water  is  discharged  under  a  head  of  64  feet  through  a  short  cylin- 
drical mouthpiece   12  in.  in  diameter.      Find  (a)  the  loss  of  head  due 
to  shock,  (b)  the  volume  of  discharge  in  cubic  feet  per  second,  (c)  the 
energy  of  the  issuing  jet.     (g  =  32.) 

Ans.  (a)  20.736  ft.;  (b)  41.23  cub.  ft. ;  (c}  201.64  H.P. 

56.  If  a  bell-mouth  is  substituted  for  the  mouthpiece  in  the  preced- 
ing question,  find  the  discharge  and  the  mechanical  effect  of  the  jet. 

Ans.  49.28  cub.  ft.  per  sec.;  344.2  H.P. 

57.  Compare  the  energies  of  a  jet  issuing  under  an  effective  head  of 
loo  ft.  through  (i)  a  12-in.  cylindrical  ajutage,  (2)  a  12-in.  divergent  aju- 
tage, (3)  a  12-in.  convergent  ajutage,  the  angle  of  convergence  being  21°. 
Draw  the  plane  of  charge  in  each  case. 

Ans.  (i)  393-8  H.P. ;  (2)  672.28  H.P.;  (3)  552.58  H.P. 

58.  Find  the  discharge  through  a  rectangular  opening  36  in.   wide 
and  10  in.  deep  in  the  vertical  face  of  a  dam,  the  upper  edge  of  the 
opening  being  10  ft.  below  the  water-surface. 

Ans.  40.2  cub.  ft.  per  sec. 

59.  A  centrifugal  pump  has  a  wheel  of  2  ft.  outside  and  i   ft.  inside 
diam.,  and  also  a  large  whirlpool  chamber.    Draw  to  scale  a  curve  show- 
ing the  pressure  at  different  points  in  the  wheel  and  whirlpool  chamber 
when  the  water  fills  the  pump  but  flows  very  slowly  towards  the  point  of 
discharge.     Take  i  atm.  as  the  pr.  at  the  inlet  surface. 

60.  A  submerged  sluice  in  the  vertical  face  of  a  reservoir  is  30  ft. 
wide.     The  effective  head  over  the  sluice  is  18  inches.     How  high  must 
the  sluice  be  raised  to  give  a  delivery  of  45,000  gal.  per  minute  ?   (c  —  .6.) 

Ans.  8.164  1MS- 

6r.  The  sill  of  a  sluice  in  the  vertical  face  of  a  reservoir  is  clear  above' 

the  tail-race ;  the  head  of  water  above  the  sill  is  5  feet.     If  the  sluice  is 

24  ft.  wide,  what  must  be  the  opening  to  give  93,750  gals,  per  min.? 

{c  =  .6.)  Ans.  12.3  ins.     • 

62.  A  sluice  in  the  vertical  side  of  a  reservoir  is  partially  submerged,. 


n6  EXAMPLES. 

the  surface  of  the  tail-race  water  being  6  ins.  above  the  sill.  The  surface 
on  the  upstream  side  is  2^  ft.  above  the  sill.  If  the  sluice  is  18  ft.  wide, 
what  must  be  the  total  opening  of  the  sluice  to  give  15,637^  tons  (of 
2000  Ibs.)  per  hour  ?  Ans.  1.203  ft*.  c  being  .6. 

63.  Find  the   discharge   in   cub.  ft.  per  sec.  through  a  sharp-edge 
orifice,  6  ins.  square,  in  a  vertical  plate,  the  centre  of  the  orifice  being  15 
ins.    below  the  water-surface,  (a)  if  the  velocity  of  approach  is  i  ft.  per 
second,  (b)  if  the  channel  of  approach  is  3  ft.  wide  by  2  ft.  deep. 

Ans.  (a)  i.  3478;  (b)  1.34. 

64.  A  reservoir  half  an  acre  in  area  with  sides  nearly  vertical,  so  that 
it  may  be  considered  prismatic,  receives  a  stream  yielding  9  cub.  ft.  per 
second,  and  discharges  through  a  sluice  4  ft.  wide,  which  is  raised  2  ft. 
Calculate  the  time  required  to  lower  the  surface  5  ft.,  the  head  over 
the  centre  of  the  sluice  when  opened  being  10  ft.  Ans.  1079  sees. 

65.  Show  that  the  energy  of  a  jet  issuing  through  a  large  rectangular 

orifice  of  breadth  B  is  i2$B(If£  —  /A*),  Hi ,  H*  being  the  depths  below 
the  water-surface  of  the  upper  and  lower  edges  of  the  orifice,  and  the 
coefficient  of  discharge  being  .625. 

66.  A  reservoir  at  full  water  has  a  depth  of  40  ft.  over  the  centre  of 
the  discharging-sluice,  which   is   rectangular  and  24  in.   wide  by  18  in. 
deep.     Find  the  discharge  in  cubic  feet  per  second  at   that  depth,  and 
-afso  when  the  water  has  fallen  to  30,  20,  and  10  ft.,  respectively  ;  find 
the  mechanical  effect  of  the  efflux  in  each  case,  c  being  .625. 

Ans.  94.8  cu.    ft.;  82.1   cu.  ft.;   67  cu.  ft.  ;  47.4  cu.    ft.;  431.2 
H.P.;  280  H. P.;  152.5  H.  P.;  53.95  H. P. 

67.  Require  the  head  necessary  to  give  7.8  cu.  ft.  per  second  through 
iin  orifice  36  sq.  in.  in  sectional  area,  c  being  .625.  Ans.  38.9  ft. 

68.  The  upper  and  lower  edges  of  a  vertical  rectangular  orifice  are 
<6  and  10  ft.   below  the  surface  of  the  water  in  a  cistern,   respectively; 
the  width  of  the  orifice  is  i  ft.      Find  the  discharge  through  it. 

Ans.  56.42  cu.  ft.  per  sec. 

69.  The  two  sluices  each  4  ft.  wide  by  2  ft.  deep  in  a  lock-gate  are 
submerged  one  half  their  depth.     The  constant  head  of  water  above  the 
-axis  of  the  sluice   is  12  ft.     Find  tlie  discharge  through  the  sluice,  the 
velocity  of  approach  being  4  ft.  per  second,  c  being  .625. 

Ans.  16,626.2  cu.  ft.  per  min. 

70.  Find  the  flow  through  a  square  opening,  one  diagonal  being  ver- 
tical and  12  in.  in  length,  the  upper  extremity  of  the  diagonal  being  .in 
the  surface  of  the  water,  and  c  being  .625.        Ans.   1.724  cu.  ft.  per  sec. 

^71.  To  find  the  quantity  of  water  conveyed  away  by  a  canal  3  ft. 
wide,  a  board  with  an  orifice  2  ft.  wide  and  i  ft.  deep  is  placed  across 
the  canal  and  dams  it  back  until  it  attains  a  height  of  2^  ft.  above  the 
bottom  and  if  ft.  above  the  lower  edge  of  the  orifice.  Find  the  dis- 
charge in  cubic  feet  per  second,  c  being  .625. 

Ans.  17.59,  or  20.21  if  orifice  is  drowned. 


EXAMPLES.  1 1 7 

72.  Six  thousand  gallons  of  water  per  minute  are  forced  through  a 
line  of  piping  A£C  and  are  discharged  into  the  atmosphere  at  C,  which 
is  6  ft.  vertically  above  A.     The  pipe  AB  is  6  in.  in  diameter  and  12  ft. 
in  length;  the  pipe  BC  is  12  in.  in  diameter  and   12  ft.  in  length.     Dis- 
regarding  friction,  find  the    "loss    in   shock"   and   draw  the   plane    of 
charge.  Ans.  Loss  of  head  in  shock  =  58.3  ft. 

73.  What  quantity  of  water  flows  through  the  vertical  aperture  of  a 
dam,  its  width  being  36  in.  and   its  depth  10  in.  ;  the  upper  edge  of  the 
aperture  is  16  ft.  below  the  surface.  Ans.  50.65.  cu.  ft.  per  sec. 

74.  264  cu.  ft.  of  water  are  discharged   through   an   orifice    of  5  sq. 
ins.  in  3  min.  10  sec.     Find  the  mean  velocity  of  efflux. 

Ans.  64  ft.  per.  sec. 

75.  One  of  the  locks  on  the  Lachine  Canal  has  a  superficial  area  of 
about  12,150  sq.  ft.,  and  the  difference  of  level  between  the  surfaces  of 
the  water  in  the  lock  and   in  the  upper  reach   is  9  ft.     Each  leaf  of  the 
gates  is  supplied  with  one  sluice,  and  the  water  is  levelled  up  in  2  min. 
48  sees.     Determine  the  proper  area  of  the  sluice-opening.     (Centre  of 
sluice  20  ft.  below  surface  of  upper  reach  and  c  =  .625.) 

Ans.  Area  of  one  sluice  =  43.39  sq.  ft. 

76.  The   horizontal  section   of  a    lock-chamber    may   be  assumed  a 
rectangle,  the  length  being  360  ft.     When  the  chamber  is  full,  the  sur- 
face width  between  the  side  walls,  which  have  each  a  batter  of  i  in  12, 
is  45  ft.      How  long  will  it  take  to  empty  the  lock  through  two  sluices  in 
the  gates,  each  8  ft.  by  2  ft.,  the  height  of  the  water  above  the  centre  of 
•the  sluices  being   13  ft.  in  the  lock  and  4  ft.  in  the  canal  on   the  down- 
stream side.  Ans.  600.75  sec.,  c  being  .625. 

77.  Water  approaches  a  rectangular  opening  2  ft.  wide  with  a  velocity 
of  4  ft.  per  second.      At  the  opening  the  head  of  water  over  the  lower 
edge  =  13  ft., and  over  the  surface  of  the  tail-race  =  12  ft.  ;  the  discharge 
through   the  opening  is  70  cu.  ft.  per  second.     Find  the  height  of  the 
opening,  c  being  .625.  Ans.  1.022  ft. 

78.  The  water  in  a  regulating-chamber  is  8  ft.  below  the  level  of  the 
water  in  the  canal  and  8  ft.  above  the  centre  of  the  discharging-sluice. 
Determine  the  rise  in  the  canal  which  will  increase  the  discharge  by  10 
percent.  Ans.  1.68  ft. 

The  horizontal  sectional  area  of  the  chamber  is  constant  and  equal  to 
400  sq.  ft.  ;  in  what  time  will  the  water  in  the  chamber  rise  to  the  level 
of  that  in  the  canal,  if  the  discharging-sluice  is  closed;  the  sluice  be- 
tween the  canal  and  chamber  being  3  sq.  ft.  in  area?  Ans.  150.83  sec. 

79.  A  lock  on  the  Lachine  Canal  is  270  ft.  long  by  45  ft.  wide  and  has 
a  lift  of  Sf  ft.  ;  there  are  two  sluices   in   each  leaf,  each  8|  ft.  wide  by 
2^  ft.  deep;   the  head  over  the  horizontal  centre  line  of  the  sluices  is 
19  ft.     Find  the  time  required  to  fill  the  lock.  Ans.  163.5  sec- 

80.  The  locks  on  the  Montgomeryshire  Canal  are  81  ft  long  and  7f 
f*.  wide;  at  one  of  the  locks  the  lift  is  7  ft.  ;  a  24-in.  pipe  leads  the  water 


Ii8  EXAMPLES. 

from  the  upper  level  and  discharges  below  the  surface  of  the  lower  level 
into  the  lock-chamber  ;  the  mouth  of  the  pipe  is  square,  2  ft.  in  the  side, 
and  gradually  changes  into  a  circular  pipe  2  ft.  in  diameter.-  Find  time 
of  filling  the  lock,  (c  =  i.)  Ans.  132.11  sec. 

81.  A   canal   lock   is   115.1    ft.  long  and   30.44  ft.  wide;    the  vertical 
depth  from  centre  of  sluice  to  lower  reach  is  1.0763  ft.,  the  charge  being 
6.3945  ft.;  the  area  of  the  two  sluices  is  2  x  6.766  sq.  ft.     Find  the  time 
of  filling  up  to  centre  of  sluices,     (c  =  .625  for  the  sluice,  but  is  reduced 
to  .548  when  both  are  opened.)     Also,  find  time  of  filling  up  to  level  of 
upper  reach  from  centre  of  sluice-doors.  Ans.  25  sec.  ;  298  sec. 

82.  How  many  gallons  of  water  will  flow  through  a  90°  notch  in  24 
hours  if  the  depth  of  the  water  is  27  ins.  for  the  first  8  hours,  12  ins.  for 
the  second  8  hours,  and  3  ins.  for  the  third  8  hours,  c  being  .6  ? 

Ans.  3,974,400. 

83.  Show  that  in  a  channel  of  V  section  an  increment  of  10  per  cent 
in  the  depth  will  produce  a  corresponding  increment  of  5  percent  in  the 
velocity  of  flow  and  of  25  per  cent  in  the  discharge. 

84.  The  angle  of   a  triangular   notch    is    90°.     How    high  must  the 
water  rise   in  the  notch  so  that  the  discharge  may  be  1000  gallons  per 
minute?  Ans.   12  ins.  very  nearly. 

85.  A  reservoir,  rectangular  in  plan  and   10,000  sq.  ft.  in  area,  has  in 
one  side  a  90°  triangular   notch   2   ft.  deep.     If  the  reservoir  is  full,  in 
what  time  will  the  level  sink  6  ins.  ?  Ans.  496.87  sees. 

86.  How  long  will  it  take  to  lower  by  3  ft.  the  surface  of  a  reservoir 
of  640,000  sq.  ft.  area  through  a  90°  V  notch  4  ft.  deep  ? 

Ans.  40.50  hrs.,  c  being  .6. 

87.  Find  the  discharge  in  cubic  feet  per  second  through  a  90°  notch 
when  the  depth  of  water  in  the  notch  is  4  ft.,  c  being  .617. 

Ans,  84.24. 

88.  A  pond  whose  area  is   12,000  sq.  ft.  has  an  overfall  outlet  36  in. 
wide,  which  at  the  commencement  of  the  discharge  has  a  head  of  2.8  ft. 

•Find  the  time  required  to  lower  the  surface  12  in.          Ans.  354.58  sec. 

'89.  How  much  water  will  flow  through  a  rectangular  notch  24111. 
wide,  the  surface  of  still  water  being '8  in.  above  the  crest  of  the  notch  ? 
(Take  into  account  side  contraction.)  Ans.  3.383  cu.  ft.  per  sec. 

'  9°*  A  weir  passes  6  cubic  feet  per  second,  and  the  head  over  the  crest 
is-8  inches.  Find  the  length  of  the  weir,  c  being  .625. 

Ans.  3.3068  ft. 

91.  A  wein  400  ft.  long,  with  a  9-111.  depth  of  water  on  it,  discharges 
through  a  lower  weir  500  ft.  long.    Find  the  depth  of  water  on  the  latter. 

Ans.  .6457  ft. 

92.  A  weir  is  545  ft.  long  ;  how  high  will  the  water  rise  over  it  when 
it  rises  .68  ft.'  upon  an  upper  weir  750  ft.  long  ?  Ans.  .8413  ft. 

;93.  What  should  be  the  height  of  a  drowned  weir  400  ft.  long,  to 
deepen  the  water  on  the  up-stream  side  by  50  per  cent,  the  section  of 


EXAMPLES.  119 

the  stream  being  400  ft.  x  8  ft.,  and  the  velocity  of  approach  3  ft.  per 
second  ?  Ans.  7.084  ft. 

*94.  The  depth  of  water  on  the  crest  of  a  rectangular  notch  5  ft.  long 
is  2  feet.     Find  the  discharge  when  the  notch  has  (a]  two  end  contrac- 
tions, (b)  one  end  contraction,  (c)  no  end  contraction,  c  in  each  case  being  f . 
Ans.  (a)    43.369   cu.    ft.    per   sec.;   (b}   45.254   cu.  ft.  per   sec.; 
(c)  47.14  cu.  ft.  per  sec. 

95.  Show  that  upon  a  weir  10  ft.  long  with  1 2  ins.  depth  of  water  flowing 
over,  an  error  of  TTTOT  °f  a  f°ot  m  measuring  the  head  will  cause  an  error 
of  3  cu.  ft.  per  minute  in  the  discharge,  and  an  error  of  T£Q  of  a  foot  in 
measuring  the  length  of  the  weir  will  cause  an  error  of  2  cu.  ft.  in  the 

•discharge. 

96.  In  the  weir  at  Killaloe  the  total  length  is  iioo  ft.,  of  which  779  ft. 
from  the  east  abutment  is  level,  while  the  remainder  slopes  i  in  214, 
giving  a  total  rise  at  the  west  abutment  of  1.5  ft.     Calculate  the  total 
discharge  over  the  weir  when  the  depth  of  water  on  the  level  part  is 
1.8   ft.,  which  gives  .3  ft.  on  highest  part  of  weir.      (Divide  slope  into 
8  lengths  of  40  ft.  each,  and  assume  them  severally  level,  with  a  head 
equal  to  the  arithmetic  mean  of  the  head  at  the  beginning  and  end  of 
each  length.)  Ans.  7496  cu.  ft.  per  sec. 

97.  A  watercourse  is  to  be  augmented  by  the  streams  and  springs 
above  its  level.     The  latter  are  severally  dammed  up  at  suitable  places 
and  a  narrow  board  is  provided  in  which  an  opening  12  in.  long  by  6  In. 
deep  is  cut  for  an  overfall ;  it  was  surmised  that  this  would  be  sufficient 
for  the  largest  streams  ;   another  piece  attached  to  the  former  would 
reduce  the  length  to  6  in.  for  smaller  streams.    Calculate  the  delivery  by 
the  following  streams  : 

In  No.  i  stream  with  the  12-in.  notch,  depth  over  crest  =  .37  ft. 
"  No.  2       "  "        "      6-in.       "  "  "         "      =  .41  ft. 

"  No.  3       "  "        "     12-in.       "  "         "      =  .29  ft. 

"  No.  4  "       "      6-in.       "  "          "         "      =.19  ft. 

(Take  into  account  the  side  contractions.) 

Ans,  No.  i,  .696cu.  ft. ;  No.  2,  .3658  cu.  ft.;  No.  3,  .4904  cu.  ft.; 
No.  4,  .1275  cu.  ft. 

98.  A  rectangular  notch  has  two  complete  end  contractions  and  the 
length  of  the  crest  is  three  times  the  depth  of  the  water  on  the  crest. 
What  must  be  the  length  of  the  crest  to  give  a  minimum  discharge  of 
18,750  gals,  per  minute,  c  being  f  ?  Ans.  5.87  ft. 

99.  A    stream    30    ft.  wide,   3    ft.    deep,  discharges    310  cu.    ft.   per 
second;  a  weir  2  ft.  deep  is  built  across  the  stream.     Find  increased 
depth  of  latter,  (a)  neglecting  velocity  of  approach,  (b)  taking  velocity  of 
approach  into  account.  Ans,  (a)  1.26  ft.  to  1.265  ft-J  (^)  I-I9  ft- 

100.  In  a  stream  50  ft.  wide  and  4  ft.  deep  water  flows  at  the  rate  of 
zoo  ft.  per  minute  ;  find  the  height  of  a  weir  which  will  increase  the  depth 


120  EXAMPLES. 

to  6  ft.,  (i)  neglecting  velocity  of  approach,  (2)  taking  velocity  of  approach? 
into  account.  Ans.  (i)  4.4126  ft.;  (2)  4.4305  ft. 

101.  A  stream  50  ft.  wide  and  4  ft.  deep  has  a  velocity  of  3  ft.  per 
second  ;  find   the  height  of  the  weir  which  will  double  the  depth,  (i) 
neglecting  velocity  of  approach,  (2)  taking  velocity  of  approach  into  ac- 
count. Ans.  (i)  5.651  ft.;  (2). 5. 6862  ft. 

102.  A  stream  80  ft.  wide  by  4  ft.  deep  discharges  across  a  vertical 
section  at  the  rate  of  640  cu.  ft.  per  second  ;  a  weir  is  built  in  the  stream, 
increasing  its  depth  to  6  ft.     Find  the  height  of  the  weir. 

Ans.  4.233  ft. 

103.  Salmon-gaps  are  constructed  in  a  weir;  they  are  each  10  ft.  wide 
and  their  crests  are  18  in.  below  the  weir  crest.     Calculate  the  discharge 
down  three  of  these  gaps,  the  water  on  the  level  part  of  the  weir  being 
8  in.  deep.  Ans.  238.15  cu.  ft.  per  sec. 

104.  A    channel    of    rectangular    section    and    20   ft.    wide    conveys 
3,600,000  gallons  per  hour,  the  depth  of  the  stream  being  8  ft.     A  dam 
2  ft.  high  is  built  across  the  channel.    Find  the  "  height  of  swell  "  (a)  dis- 
regarding the  velocity  of  approach,  (b)  taking  the  velocity  of  approach 
into  account.  Ans.  (a)  .07  ft.;  (b)  .0545  ft. 

105.  The  water  in  a  flume  8  ft.  wide  is  3  ft.  deep  and  is  supplied  from 
a  sluice  6  ft.  wide  at  the  rate  of  27,000  gals,  per  minute.     If  the  coeffi- 
cient of  contraction  is  unity  and  if   10  per  cent  is  allowed  for  fractional 
loss,  find  the  difference  of  level  between  the  water-surfaces  above  the 
sluice  and  in  the  flume  when  the  sluice  opening  is  (a)  i  ft.,  (b)  2  ft. 

Ans.  (a)  2.32  ft.;  (b)  .31  ft. 

106.  A  stream  of  rectangular  section  24  ft.  wide  delivers  145  cu.  ft. 
per  second.     The  edge  of  a  drowned  weir  is  15  ins.  below  the  surface  of 
the  water  on  the  down-stream  side.     Determine  the  difference  of  level 
between  the  surfaces  of  the  water  on  the  up-  and  down-stream  sides,  the 
velocity  of  approach  being  2  ft.  per  second. 

Ans.  7.9  ins. 


CHAPTER    II. 


FLUID    FRICTION   AND    PIPE    FLOW. 

I.  Fluid  Friction. — The  term  fluid  friction  is  applied  to 
the  resistance  to  motion  which  is  developed  when  a  fluid  flows 
over  a  solid  surface,  and  is  due  to  the  viscosity  of  the  fluid. 
This  resistance  is  necessarily  accompanied  by  a  loss  of  energy 
caused  by  the  production  of  eddies  along  the  surface,  and 
similar  to  the  loss  which  occurs  at  an  abrupt  change  of  section, 
or  at  an  angle  in  a  pipe  or  channel. 

Froude's  experiments  on  the  resistance  to  the  edgewise 
motion  of  planks  in  a  fluid  mass,  the  planks  being  TSF  in.  thick,. 
19  in.  deep,  and  I  to  50  ft.  long,  each  plank  having  a  fine 
cutwater  and  run,  are  summarized  in  the  following  table: 


Length  of  Surface  in  Feet. 

Nature  of  Surface 

2  Feet. 

8  Feet. 

20  Feet. 

50  Feet. 

Covering. 

A 

B 

C 

A 

B 

C 

A 

B 

C 

A 

B 

C 

Varnish   

2  .OO 

.  4  1 

.  3QO 

i  S* 

q2^ 

.26.1 

i  S^ 

278 

o  in 

r  81 

2Cf> 

Paraffine 

oft 

070 

I    Q-l 

O  T  A 

260 

27  I 

Tinfoil  

">  16 

2QC 

I  .  QQ 

278 

26  "\ 

L  •yj 

I    OO 

262 

•"O/ 
2J/1 

r   S-; 

?  16 

Calico  

1-93 

.87 

•725 

1.92 

.626 

•5°4 

1.89 

•5.31 

•447 

I.&7 

•  47-1 

.232- 
•423. 

Fine  sand  

2.OO 

.81 

.  6go  2  .  oo 

.583 

•450 

2.00 

.480 

.38.4 

2.O(J 

.40^ 

•337 

Medium  sand  

2.0O 

.go 

•  730 

2.OO 

.625 

.488 

2.OO 

•F.34 

.46*; 

2.0O 

.488 

.456. 

Coarse  sand  

2.0O 

I.  IO 

.880 

2.  CO 

•714 

.520 

2.OO 

.588 

.490 

Columns  A  give  the  power  of  the  speed  (v)  to  which  the 
resistance  is  approximately  proportional. 

Columns  B  give  the  mean  resistance,  in  pounds  per  square 

121 


122  FLUID  FRICTION. 

foot,  of  the  whole  surface  of  a  board  of  the  lengths  stated  in 
the  table.' 

Columns  C  give  the  resistance,  in  pounds,  of  a  square  foot 
of  surface  at  the  distance  sternward  *V0m  the  cutwater  stated 
in  the  heading,  each  plank  having  a  standard  speed  of  10  ft. 
per  second.  The  resistance  at  other  speeds  can  be  easily  cal- 
culated. 

An  examination  of  the  table  shows  that  the  mean  resistance 
per  square  foot  diminishes  as  the  length  of  the  plank  increases. 
This  may  be  explained  by  the  supposition  that  the  friction  in 
the  forward  portion  of  the  plank  develops  a  force  which  drags 
the  water  along  with  the  surface,  so  that  the  relative  velocity 
of  flow  over  the  rear  portion  is  diminished.  Again,  the 
-decrease  of  the  mean  resistance  per  square  foot  is  .132  Ib.  when 
the  length  of  a  varnished  plank  is  increased  from  2  to  20  ft. , 
while  it  is  only  .028  Ib.  when  the  length  increases  from  20  to 
50  ft.  Hence  for  greater  lengths  than  50  ft.  the  decrease  of 
resistance  may  be  disregarded  without  much,  if  any,  practical 
effect. 

Thus,  generally  speaking,  these  experiments  indicate  that 
the  mean  resistance  is  proportional  to  the  nth  power  of  the 
relative  velocity,  ;/  varying  from  1.83  to  2.16,  and  its  average 
value  being  very  nearly  2. 

Colonel  Beaufoy,  as  a  result  of  experiments  at  Deptford, 
also  assumed  the  mean  resistance  to  be  proportional  to  the 
?/th  power  of  the  relative  velocity,  the  value  of  n  in  three  series 
of  observations  being  1.66,  1.71,  and  1.9. 

The  frictional  resistance  is  evidently  proportional  to  some 
function  of  the  velocity,  F(?'),  which  should  vanish  when  v  is 
nil,  as  when  the  surface  is  level,  and  should  increase  with  v. 

Coulomb  assumed  the  function  /7(r)  to  be  of  the  form 
av  -\-  biP,  -a  and  b  being  coefficients  to  be  determined  by  ex- 
periment. Experiment  shows  that  when  b  does  not  exceed 
5  ft.  per  minute  the  resistance  is  directly  proportional  to  the 
velocity,  but  that  ic  is  more  nearly  proportional  to  the  square 


FLUID  FRICTION.  123 

of  the  velocity  When  the   velocity  exceeds   30  ft.  per  minute ; 
<5r, 

F(v).  =  av  when  sv  ~     5  ft.,  per  minute, 
and 

F(v)  =  fo'2  when  ^  ~  30  ft.  per  minute. 

Again,  observations  on  the  flow  of  water  in  town  mains 
indicate  that  no  difference  of  resistance  is  developed  under 
widely  varying  pressures,  and  this  independence  of  pressure  is 
also  verified  by  Coulomb's  experiment  showing  that,  if  a  disc 
is  oscillated  in  water,  there  is  no  apparent  change  in  the  rate 
•of  decrease  of  the  oscillations,  whether  the  water  is  under 
atmospheric  pressure  or  not. 

From  the  preceding  and  other  similar  experiments  the  fol- 
lowing general  laws  of  fluid  friction  have  been  formulated: 

(1)  The  frictional  resistance  is  independent  of  the  pressure 
between  the  fluid  and  the  surface  over  which  it  flows. 

(2)  The  frictional   resistance  is  proportional  to  the  are£  of 
the  surface. 

(3)  The  frictional  resistance  is  proportional  to  some  func- 
tion, usually  the  square,  of  the  velocity. 

To  these  three  laws  may  be  added  a  fourth,  viz.  : 

(4)  The  frictional  resistance  is  proportional  to  the  density 
and  viscosity  of  the  fluid. 

A  fifth  law,  viz.,  that  "the  frictional  resistance  is  inde- 
pendent of  the  nature  of  the  surface  against  which  the  fluid 
flows,"  has  been  sometimes  enunciated,  and  at  very  low 
velocities  the  law  is  approximately  true.  At  high  velocities, 
however,  such  as  are  common  in  engineering  practice,  the 
resistance  has  been  shown  by  experiment,  and  especially  by 
the  experiments  carried  out  by  Darcy,  to  be  very  largely 
influenced  by  the  nature  of  the  surface. 


124  FLUID  FRICTION. 

Let  p  be  the  (fictional  resistance  in  pounds  per  square  foot 
of  surface  at  a  velocity  of  I  ft.  per  second. 

Let  A  be  the  area  of  the  surface  in  square  feet. 

Let  i'  be  the  relative  velocity  of  the  surface  and  the  water 
in  which  it  is  immersed. 

Let  R  be  the  total  frictional  resistance. 

Then  from  the  laws  of  fluid  friction 

R  =  p  .  Av\ 

<2<r 

Take  f  =.  ----  />,  w  being  the  specific  weight  of  the  fluid*, 
Then 

R  =fwA  —  . 

*g 

The  coefficient  f  is  approximately  constant  for  any  given 
surface,  and  is  termed  the  coefficient  of  fluid  friction.  The 
power  absorbed  by  the  frictional  resistance 


. 

<b 

TACLE  GIVING  THE  AVERAGE  VALUES  OF  /  IN  THE  CASE  OF 
LARGE  SURFACES  MOVING  IN  AN  INDEFINITELY  LARGE 
MASS  OF  WATER. 

Surface.  x  Coefficient  of  Friction  (/").. 

New  well-painted  iron  plate  ..........  ,    .00489 

Painted  and  planed  plank  ..............  °°35 

Surface  of  iron  ships  ..................  00362 

Varnished  surface  ....................  00258 

Fine  sand  surface  ..........  %  ..........  004  1  8 

Coarse  sand  surface  ...................  00503 

Ex.  The  wetted  surface  of  a  vessel  moving  at  8  knots  per  hour  is. 
7500  sq.  ft.,  and  the  resistance  is  .4  Ibs.  per  sq.  ft.  at  a  speed  of  10  ft.  per 
second.  Find  the  surface-resistance  and  the  horse-power  required  to 
propel  the  vessel. 

The  resistance  in  Ibs.  per  sq.  ft.  at  i  ft.  per.  sec.  =  —  -,  ==  -  •. 

I02          JOOO 


FLUID  FRICTION,  125 

Therefore  the  total  skin-resistance 

4  /8  x  6086  \9 

-  -7500-2          z—     —  5487-3  lbs- 
1000    '        \6o  x  60  y 

The  horse-power  =  5487.3   x   --  -r  —    -  x  -  =  134.93. 

2.  Surface  Friction  of  Pipes.  —  Assuming  that  the  laws  of 
fluid  friction  already  enunciated  hold  good  when  water  flows 
through  a  pipe,  it  has  been  shown  by  numerous  experiments 
that  the  coefficient  of  friction  f  lies  between  the  limits  .005  and 
.OI,  its  average  value  under  ordinary  conditions  being  about 
.0075.  No  single  value  of  f  is  applicable  to  very  different 
cases.  Indeed,  /depends  not  only  upon  the  condition  of  the 
surface,  but  also  upon  the  diameter  of  the  pipe  and  the  velocity 
of  the  water.  Some  authorities  have  expressed  its  value  by  a 
relation  of  the  form 

/  6 


a  and  b  being  constants  whose  values  are  to  be  determined  by 
experiment. 

The    following    table    gives    some   of   the    best    numerical 
results  obtained  for  a  and  b  :  ' 

Authority.  Q  b 

Prony  ..........  .......  0002  1230  .00003466 

D'  Aubuisson  ..........  0002090  .000037608 

Eytelwein  .......  ......  00017059  .00004441 

In  pipes  of  small  diameter  in  which  the  velocity  of  flow  is 

less  than  4  ins.  per  second  the  term  a  may  be  disregarded  so 

that 


In  ordinary  practice  and  when  the  pipes  have  been  in  use 


126 


SURFACE  FRICTION  OF  PIPES. 


for  some  time  the  velocity  usually  exceeds  4  ins.  per  second* 
and  the  term  —  may  then  be  disregarded,  so  that 


s 

Darcy's  and  other  more  recent  experiments  show  that  a 
and  b  are  not  constant,  but  are  more  correctly  expressed  as. 
functions  of  the  diameter.  In  Darcy's  experiments  the  pipes 
were  laid  very  nearly  horizontal  and  the  head  could  be  varied 
at  will  by  the  opening  or  closing  of  valves. 


-164- 


at-  •- 


FIG.  79. 

Piezometers  were  inserted  at  intervals  of  164  ft.  (50  m.)> 
commencing  at  15.4  ft.  (4.7  m.)  from  the  inlet,  i.e.,  at  a, 
point  where  the  pipe  was  running  full  and  the  flow  was  steady. 
The  upper  ends  of  the  piezometers  terminated  on  a  vertical 
plank  so  placed  as  to  allow  the  water-levels  in  them  to  be 
observed  and  compared.  In  any  two  consecutive  piezometers 
the  difference  of  level,  which  is  of  course  constant,  represents, 
the  frictional  loss  of  head  in  a  i64-ft.  length  of  pipe.  From 
the  results  of  these  experiments  Darcy  made  the  following 
deductions : 

(a)  The  frictional  resistance  depends  upon  the  material  ancf 
condition  of  the  pipe. 

For  example,  the  resistance  to  flow  is  much  less  in  a  glass 
than  in   an  iron  pipe,  and  is  approximately  twice  as  great  in 


DARCY'S  EXPERIMENTAL   RESULTS.  127 

pipes  which  have  become  incrusted  with  use  as  in  new  clean 
pipes.  It  must  be  remembered,  however,  that  although  numer- 
ous experiments  have  been  made  with  new  pipes,  there  have  been 
comparatively  few  experiments  with  old  pipes.  Thus,  in  pipes 
in  which  the  velocity  of  flow  exceeds  4  ins.  per  second,  Darcy 

considered  it  more  correct  to  express  a  ( =  -}  in  the  form 

\      g' 

I— •+* 

d  being  the  diameter  of  the  pipe,  and  a  and  ft  coefficients  to  be 
determined  by  experiment.  The  following  values  for  a  and  ft 
are  given  by  Darcy: 

ft 
For   drawn    wrought-iron    or    smooth 

cast-iron  pipes 00x31545      .000012973 

For    pipes    with    surface    covered    by 

light  incrustations .0003093      .00002598 

Without  sensibly  altering  the  values  of  these  coefficients 
they  can  be  put  into  the  following  simple  form : 


?—;(•+=>• 


d  being  the  diameter  in  feet,  and  ^  being  .005  or  .01  according 
as  the  pipes  are  clean  or  haVe  become  slightly  incrusted. 

(b)  The  coefficient  b  is  not  constant,  but  varies  slightly  both 
with  the  diameter  and  the  velocity ',  its  value  diminishing  as  d 
or  v  increases. 

In  practice  it  is  assumed  that  b  is  constant  and  the  error 
involved  has  the  advantage  of  giving  to  the  pipe  a  larger  sec- 
tional area  than  is  actually  required  for  a  given  discharge. 
Thus  allowance  is  partially  made  for  the  incrustations  with 
which  the  surface  gradually  becomes  covered. 


128 


DARCY'S  EXPERIMENTAL   RESULTS. 


Darcy  proposed  to  include  all  cases  in  the  more  general 
form 


=  +K  +  -; -- 


in  which,  for  new  and  smooth  iron  pipes, 


a  =  .00001350 
a'  =  .000031635 


ft  =  .000012402 
ft'  —  .00000016186 


This  value  for  —  is  rarely  if  ever  used. 

TABLE   GIVING    DARCY'S    VALUES    OF  /  FOR   VELOCITIES 
EXCEEDING  4  IN.   PER    SECOND. 


Diam. 

Value  of  f. 

Diam. 

Value  of/. 

Diam. 

Value  of/. 

of 

of 

of 

Pipe 

Pipe 

Pipe 

in 

New 

Incrusted 

in 

New 

Incrusted 

in 

New 

Incrusted 

Inches. 

Pipes. 

Pipes. 

Inches. 

Pipes. 

Pipes. 

Inches. 

Pipes. 

Pipes. 

2 

.0075 

.0150 

Q 

.00556 

.OIIII 

27 

.00519 

.01037 

3 

.00667 

•01333 

12 

.00542 

.01083 

30 

.00517 

.01033 

4 

.00625 

.0125 

15 

•00533 

.01067 

36 

.00514 

.OIO28 

5 

.0060 

.OI2 

18 

.00528 

.01056 

42 

.OO5I2 

.OIO24 

6 

.00583 

.01167 

21 

.00524 

.01048 

48 

.00510 

.01021 

7 

.00571 

.01.143 

24 

.00521 

.OIO42 

54 

.00509 

.01019 

8 

-00563 

.OII25 

Weisbach  gives  the  formula 


/=  .0036 


,00429 


Poiseuille's  experiments  indicate  that  the  surface  friction  in 
capillary  tubes  is  directly  proportional  to  the  velocity,  but  in 
pipes,  in  ordinary  practice,  the  frictional  resistance  is  certainly 
more  nearly  proportional  to  the  square  of  the  velocity,  and 
must  be  largely  due  to  eddies  which  are  the  more  readily 
formed  as  the  viscosity  diminishes.  This  viscosity,  again, 


POISEUILLE'S  EXPERIMENTS.  129 

increases  as  the  temperature  falls,  and  the  surface  friction  is 
diminished  by  about  I  per  cent  for  every  rise  of  5°  F.  in  the 
temperature.  The  resistance  to  the  motion  of  a  body  in  water, 
or  to  the  flow  of  water  along  a  surface,  is  evidently  of  two 
lands,  the  one  due  to  surface  contact,  the  other  to  the  forma- 
tion of  eddies.  Hele  Shaw's  experiments  clearly  show  the 
effect  of  surface  contact  upon  stream-line  motion  and  the 
manner  in  which  the  motion  is  modified  by  the  presence  of 
obstacles  (Trans.  Naval  Architects,  1897-98),  while  the  two 
kinds  of  resistance  are  plainly  demonstrated  by  the  interesting 
experiments  of  Osborne  Reynolds.  The  water  flows  through 


FIG.  80. 

a  glass  pipe  AB  having  a  trumpet-shaped  mouth  A.  A  glass 
tube  CD  with  a  funnel  E  terminates  in  a  pipette  Fy  the  axis 
of  the  pipette  being  in  line  with  the  axis  of  the  pipe.  The  tube 
is  filled  with  an  aniline  dye  which  is  allowed  to  escape  through 
the  pipette  in  a  thin  thread-like  stream,  the  discharge  being 
governed  by  a  small  cock.  So  long  as  the  velocity  of  flow  in 
the  pipe  does  not  exceed  a  certain  value,  which  Reynolds  calls 
the  critical  velocity,  the  aniline  thread  is  unbroken,  so  that  the 
motion  of  the  water  is  undisturbed  and  must  be  in  parallel 
lines.  As  soon  as  the  critical  velocity  is  exceeded  the  colored 
thread  is  broken  up,  becoming  sinuous  in  character,  and  the 
parallel  stream-line  motion  is  completely  destroyed  within  a 
very  short  distance  from  the  mouth  of  the  pipe. 

According  to  Reynolds  the  critical  velocity  (z/tf),  in  metres 
per  sec.,  is  given  by  the  formula 


130  FLUID  FRICTION. 

jj.  being    (?)  for  capillary  tubes  and  --•-=  for  ordinary  pipes,, 

while 

-73  =  i  -j-  .O336/  -[-  .00022 it2, 

t  being  the  temperature  in  degrees  centigrade. 

It  has  been  shown  by  H.  T.  Barnes,  D.Sc.,  in  his  experi- 
ments on  the  specific  heat  of  water,  that,  if  water  be  heated 
while  flowing  through  a  tube  at  velocities  less  than  the  critical 
velocity,  the  temperature  distribution  in  the  column  is  not 
uniform.  If  the  heat  be  applied  electrically,  by  means  of  a 
wire  threaded  through  the  flow-tube,  the  hot  water  flows  along 
the  wire,  leaving  the  walls  of  the  tube  almost  entirely  unheated. 
If  the  heat  be  applied  to  the  walls  of  the  tube,  the  colder 
water  passes  through  the  centre  of  the  tube  unheated,  leaving 
a  cloak  of  hot  water  along  the  sides.  In  neither  case  is  there 
any  tendency  to  mix  as  long  as  stream-line  flow  is  maintained. 

A  new  method  for  determining  the  critical  velocity  of  a 
fluid,  based  on  the  above  experiments,  has  been  recently 
worked  out  by  Drs.  Barnes  and  Coker  in  the  McGill  hydraulic 
laboratory.  In  this  method,  a  sensitive  mercury  thermometer 
is  placed  exactly  in  the  centre  of  a  column  of  water  as  it 
emerges  from  the  tube  under  examination,  with  the  bulb  just 
beyond  the  end.  The  walls  of  the  tube  are  maintained  at  a 
constant  temperature,  slightly  above  that  of  the  water  flowing 
'through,  but  for  stream-line  flow  the  temperature  indicated  by 
the  thermometer  will  be  that  of  the  water  in  the  head  supplying 
the  constant  flow.  The  arrival  of  the  critical  velocity,  at  which 
stream-line  flow  becomes  eddying  and  sinuous,  is  at  once  shown 
by  a  sudden  small  increase  in  the  reading  of  the  thermometer, 
and  is  due  to  the  mixture  of  the  water-film  next  the  surface 
with  the  colder  water  flowing  through  the  body  of  the  pipe. 
The  point  is  very  sharply  defined,  and  the  method  is  in  many 
cases  far  more  applicable  and  convenient  than  the  usual  color- 
band  test. 


SHIP  RESISTANCE.  131 

The  experiments  now  in  progress  in  the  hydraulic  labora- 
tory by  Barnes  and  Coker  are  being  made,  both  by  the 
thermal  and  color-band  methods,  under  the  most  favorable 
conditions  for  securing  the  perfectly  steady  conditions  neces- 
sary for  maintaining  stream-line  flow.  The  results  so  far 
obtained  show  that  the  effect  of  temperature  is  very  marked  in 
altering  the  point  of  instability  of  flow,  and  that  this  variation 
accords  at  least  approximately  with  the  formula  quoted  by 
Osborne  Reynolds  and  taken  from  Poiseuille's  experiments. 
The  effect  of  pressure  has  been  studied  over  a  limited  range,, 
and  it  has  been  shown  that  water  flowing  under  a  high  head 
has  greater  stability,  which  means  that  there  is  a  definite 
increase  in  the  velocity  at  which  stream-line  motion  breaks 
down.  Indeed,  under  the  present  arrangements,  it  has  been 
possible  to  maintain  stream-line  motion  to  very  much  higher 
velocities  than  is  possible  in  experiments  carried  out  with  the 
apparatus  used  by  Reynolds. 

3.  Resistance  of  Ships. — The  motion  of  a  ship  through 
water  causes  the  production  of  waves  and  eddies,  and  the  total 
resistance  to  the  movement  of  a  ship  is  made  up  of  a  frictional 
resistance,  a  wave-making  resistance,  and  an  eddy-making 
resistance.  Although  there  is  no  theory  'by  which  the  resist- 
ance at  a  given  speed  of  a  ship  of  definite  design  can  be 
absolutely  determined,  Froude's  experiments  render  it  possible 
to  make  certain  inferences  and  furnish  some  useful  data. 

According  to  Froude,  the  frictional  resistance'  is  sensibly 
the  same  as  that  of  a  rectangular  surface  moving  with  the  same 
speed,  of  the  same  length  as  the  ship  in  the  direction  of  motion, 
and  of  an  area  equal  to  the  immersed  surface  of  the  ship. 
Experiments  seem  to  indicate  that  as  the  speed  increases,  the 
frictional  resistance  of  well-designed  ships  with  clean  bottoms 
is  from  90  to  60  per  cent  of  the  total  resistance,  and  that  the 
percentage  is  greater  when  the  bottoms  become  foul. 

The  wave-making  resistance  is  especially  affected  by  the 
form  and  proportions  of  the  ship,  depending,  for  a  given 


132  SHIP   RESISTANCE. 

length,  upon  the  proportions  of  the  entrance,  middle  body,  and 
run.  For  every  ship  there  is  a  limit  of  speed  below  which  the 
resistance  is  approximately  proportional  to  the  square  of  the 
speed,  being  chiefly  due  to  friction,  and  beyond  which  it 
increases  more  rapidly  than  as  the  square. 

The  eddy-resistance  in  the  case  of  well-formed  ships  should 
not  exceed  about  10  per  cent  of  the  total  resistance,  and  is  often 
much  less. 

Froude's  law  of  resistance  may  be  enunciated  as  follows: 

Let  /t ,  /2  be  the  lengths  of  a  ship  and  its  model. 

Let  A^ ,  A^  be  the  displacements  of  a  ship  and  its  model. 

Let  R^ ,  R2  be  the  resistances  of  a  ship  and  its  model  at  the 
speeds  ?/t  and  vy 

Then,  if 


the  resistances  are  in  the  ratio  of 

*«  _£.  _A1 

R     ~  A^~  I* 

Hence,  too,  the  H.P. ,  and  therefore  ai so  the  coal  consumption 
per  hour,  is  proportional  to  Rv,  that  is,  to 

.7  ,7 

A*     or     /?     or     ?'', 

^.nd  the  coal  consumption  per  mile  is  proportional  to 
A     or     /3     or     ?'6. 

Again,  ^  is  proportional  to  /3; 
that  is,  to  /X  /2; 

that  is,  to  z/2  X  ^f ; 

and   it  is  sometimes  convenient  to  express  •  the    resistance  in 
pounds  in  the  form 

R  =  k. 


PIPE  FLOW  ASSUMPTIONS.  133 

•u  being  the  speed  in  knots,  A  the  displacement  in  tons,  and  k 
a  coefficient  depending  upon  the  type  of  ship  and  varying  from 
.55  to  .85  when  the  bottom  is  clean. 

Ex.  If  the  New  York,  with  a  displacement  of  10,000  tons  and  requir- 
ing 20.000  H.  P.  for  a  speed  of  20  knots,  is  taken  as  the  model  for  a  new 
steamer  which  is  to  have  a  speed  of  21  knots,  then 

/2I\e 

new  steamer's  displacement  =  10,000  (  —  j  —  13,400,  approximately, 

/2I\7 

H.  P.  of  new,  steamer  =  2o,ooof  —  J  =  28,000,  approximately. 

4.  Pipe-flow  Assumptions. — In  the  ordinary  theory  of  the 
flow  of  water  in  a  pipe  it  is  assumed  that  the  water  consists  of 
thin  plane  layers  perpendicular  to  the  axis  of  the  pipe,  that 
each   layer  is  driven  through  the  pipe  by  the  action  of  gravity 
and  by  the  difference  of  pressure  on  its  plane  faces,  and  that 
the  liquid  molecules  in  any  layer  at  any  given  moment  will  also 
be  found  in  a  plane  layer  after  any  interval  of  time.      In  such 
motion  the  internal  work  done  in  deforming  a  layer  may  be 
generally  disregarded. 

It  is  further  assumed  that  there  is  no  variation  of  velocity 
over  the  surface  of  a  layer,  and  this  is  equivalent  to  saying  that 
each  liquid  molecule  in  a  cross-section  has  the  same  mean 
velocity. 

The  disagreement  of  these  assumptions  with  the  results  of 
recent  experimental  researches  will  be  referred  to  in  a  subse- 
quent article. 

5.  Steady  Motion  in  a  Pipe  of  Uniform  Section. — Since 
the  motion  is  to  be  steady,  the  same  volume  Q  cu.  ft.  of  water 
will  always  arrive  at  any  given  cross-section  of  A  sq.  ft.  with 
the  same  mean  velocity  v  ft.  per  second.      Then 

Q  =  Av. 

But  since  the  pipe  is  of  constant  diameter,  A  is  constant,  and 
hence  also  v  is  constant,  so  that  the  mean  velocity  is  the  same 
throughout  the  whole  length  of  the  pipe. 


FLOW  IN  PIPE   OF  UNIFORM  SECTION. 


Consider  an  elementary  mass  of  the  fluid  A  ABB,  bounded 
by  the  pipe  and  by  the  two  .cross-sections  A  A  ,  BB.  Let  dt, 
be  the  length  AB  of  the  ele- 
ment, the  length  /  ft.  of  the 
pipe  being  measured  along  the 
axis  from  any  origin  O. 

Let  z,  s  -f-  dz  be  the  eleva- 
tions in  feet  above  a  datum  line 
of  the  centres  of  pressure  in  the 
cross-sections  AA,  BB,  respec- 
tively. 

Let/,  p  -{-  dp  be  the  intensi- 
ties of  the  pressures  on  these  FIG.  81. 

cross-sections  in  pounds  per  square  foot. 

Let  P  be  the  perimeter  of  the  pipe. 

Let  w  be  the  specific  weight  of  the  water  in  pounds  per 
cubic  foot. 

Work  Done  by  Gravity.  —  In  one  second  wQ  Ibs.  of  water 
are  transferred  from  A  A  to  BB,  falling  through  a  vertical  dis- 
tance of  dz  ft.  Thus  the  work  done  by  gravity  per  second 


a  positive  quantity  if  dz  is  negative,  and  vice  versa. 

Work  Done  by  Pressure.  —  The  total  pressure  on  A  A  paral- 
lel to  the  axis  =  pA  ;  the  total  pressure  on  BB  parallel  to  the 
axis  =  (p  +  dp)A. 

Therefore  the  total  resultant  pressure  parallel  to  the  axis 
in  the  direction  of  motion  =  —  A  .  dp,  and  the  work  done  per 
second  on  the  volume  Q  by  this  pressure  =  —  Q  .  dp. 

NOTE.  —  The  work  done  by  the  pressure  at  the  pipe  surface  is  nil,  as 
Its  direction  is  at  right  angles  to  the  line  of  motion. 

Work  Absorbed  by  Frictional  Resistance.  —  From  the  laws 
of  fluid  friction  this  work  per  second  is  evidently 

.      =  _  p  .  dl  .  F(v)  X  v  =  -  -j-  .  Q  .  F(v)  •  dlt 


FLOW  IN  PIPE  OF  UNIFORM  SECTION.  135 

the  sign  being  negative  as  the  work  is  done  against  a  resist- 
ance. 

*  *."  . 

Since  the  motion  is  steady,  the  work  done  by  the  external 
forces  must  be  equivalent  to  the  work  absorbed  by  the  frictional 
esistance,  and  hence 

P 

—  wQ.dz  —  Q  .  dp  —  -^  Q  .  F(v)  .  dl  =  o, 

or 

*+*  +  4;:S2U  =  a 

w         A        w 

Integrating, 

p     '  P    F(v) 
z  -\-  —  A-  -r  .  -—  .  /  =  a  constant  =  H, 

1    w    '   A        w 

so  that  /fft.-lbs.  per  pound  of  fluid  is  the  uniformly  distributed 
total  constant  energy. 

A 

-p  is  called  the  hydraulic  mean  radius  of  a  pipe  and  will  be 

denoted  by  in. 
Take 


W 


the  value  adopted  in   ordinary  practice,  /being  the  coefficient 
of  friction.      Then 

P       fl  v2 
z+w  +  £^  = 

Let  zl  ,  Al,^>l  be  the  elevation  above  datum,  the  area  of 
the  cross-section,  and  the  intensity  of  the 
pressure  at  any  point  X  on  the  axis  of 
the  pipe  distant  /x  from  the  origin  (Fig. 

82). 

Let  ,sr2,  y42,/2be  the  elevation  above  datum,  the  area  of 
the  cross-section,  and  the  intensity  of  the 
pressure  at  any  other  point  Y  on  the  axis 
distant  /2  from  the  origin  (Fig.  82). 


I36  FLOW  IN  PIPE  OF  UNIFORM  SECTION. 

Then,  from  the  equation  just  deduced, 


m  2g 


Hence 


-.+9- 


w         m  2g 
f   v\,  fL    -v* 


m 


m 


L  being  the  length  /2  —  ^  of  the  pipe  between  the  two  points 
K 


FIG.  82. 


Let  vertical  tubes  (pressure-columns)  be  inserted  in  the 
pipe  at  X  and  at  K  The  water  will  rise  in  these  tubes  to  the 
levels  C  and  D,  and  evidently 


^  being  the  intensity  of  the  atmospheric  pressure. 


FLOW  IN  PIPE  OF  UNIFORM  SECTION.  137 

Hence,  if  CX  and   DY  are  produced  to  meet   the  datum 
lirie  in  E  and  F, 


and 


Therefore 


=  CE  + 

w  rw 


=  DF  +     L. 

w  w  w 


\ 


m 


G  being  the  point  in  which  the  horizontal  through  C  meets  FD 
produced. 

DG  is  called  the  "virtual  fall  "  of  the  pipe,  being  the  fall 
of  level  in  the  pressure-columns ;  and  since  there  would  be  no 
fall  of  level  if  the  friction  were  nil,  DG  is  said  to  be  the  head 
lost  in  friction  in  the  distance  XY. 

Denote  this  head  by  h ;  then 

m  zg'j 
and  therefore 

h          f   v* 
L  ~  '   m  2g* 

This  ratio  -j  is  designated  the  virtual  slope  of  the  pipe, 

and  is  the  head  lost  in  friction  per  unit  of  length.  It  will  be 
denoted  by  2,  so  that 

h.  f     V2 

L  "        "  mT  2g* 

If  the  section  of  the  pipe  is  a  circle  of  diameter  d,  or  a 
square  with  a  side  of  length  d,  then 

_A        d.         £ 
™~P=  4  >  "   *- 


I38  INFLUENCE  OF  PIPE'S  INCLINATION. 

and 

h         .  _  4/  v*          v^ 
L  (f^  2g  r ' 

where   a  =  —  and  r  is  the  radius. 

g 
6.  Influence  of  the  Pipe's  Position  and  Inclination  on  the 

Flow. — In  Fig.  82  join  CD.  Now  since  the  fall  of  level  (h} 
is  proportional  to  L,  the  free  surface  in  any  other  column 
between  X  and  Y  must  also  be  on  the  line  CD.  Thus  the 
pressure  /'  at  any  intermediate  point  M  distant  x  (=  XM) 
from  X  is  given  by 

»-  =  MN  +  --  =  CX+  -^(DY  —  CX\  +  ^. 

w  w  />  w 

Hence,  at  every  point  of  a  pipe  laid  below  CD,  the  fluid 
pressure  (/')  exce'eds  the  atmospheric  pressure  (/0)  by  an 
amount  w  .  MN,  so  that  if  holes  are  made  in  such  a  pipe,  the 
water  will  flow  out  and  there  will  be  no  tendency  on  the  part 
of  the  air  to  flow  in.  In  pipes  so  placed  vertical  bends  may 
be  introduced,  care  being  taken  to  provide  for  the  removal  of 
the  air  which  may  collect  in  the  upper  parts  of  the  bends. 

If  the  line  of  the  pipe  coincides  with  CD,  i.e.,  with  the 
virtual  slope  or  line  of  free  surface  level,  MN  •=.  o,  and  the 
fluid  pressure  is  equal  to  that  of  the  atmosphere.  If  holes  are 
now  made  in  the  pipe,  it  can  easily  be  shown  by  experiment 
that  there  will  be  neither  any  tendency  on  the  part  of  the  water 
to  flow  out  nor  on  the  part  of  the  air  to  flow  in. 

Next  take  CC'  =  DD'  =  ^,  and  join  C'D'. 

w 

If  the  pipe  is  placed  in  any  position  between  CD  and  CfD't 
MN  becomes  negative,  and  the  fluid  pressure  in  the  pipe  is  less 
than  that  of  the  atmosphere.  If  holes  are  made  in  this  pipe^ 
there  will  be  no  tendency  on  the  part  of  the  water  to  flow  out, 
but  the  air  will  flow  in.  Thus,  if  a  pipe  rises  above  the  line 


PIPE  FORMUL/E. 


139 


of  virtual  slope,  there  is  a  danger  of  air  accumulating  in  the 
pipe  and  impeding,  or  perhaps  wholly  stopping,  the  flow. 
No  vertical  bends  should  be  introduced,  as  the  air  is  easily  set 
free  and  would  collect  in  the  upper  parts  of  the  bends,  with 
the  effect  of  impeding  the  flow  and  of  acting  detrimentally 
upon  the  water  itself,  which  the  liberation  of  the  air  renders 
less  wholesome.  If  the  line  of  pipe  coincides  with  C'D',  then 
the  fluid  pressure  is  nil. 

Finally,  if  the  pipe  at  any  point  rises  above  C'D't  the 
pressure  becomes  negative,  which  is  impossible.  In  fact,  the 
continuity  of  flow  is  destroyed,  and  the  pipe  will  no  longer 
run  full  bore.  Air  will  be  disengaged  and  will  rise  and  collect 
at  the  point  in  question,  so  that  in  order  to  prevent  the  flow 
being  wholly  impeded,  it  will  be  necessary  to  introduce  an  air- 
chamber  at  this  point  from  which  the  air  can  be  removed  when 
required. 

NOTE. — In  the  preceding  it  has  been  assumed  that  the  pipe  is 
straight.  If  the  pipe  is  curved,  so  also  is  the  line  of  virtual  slope.  In 
ordinary  practice,  however,  the  vertical  changes  of  level  in  a  pipe  at 
different  points  are  small  as  compared  with  the  length  of  the  pipe,  and 
distances  measured  along  the  pipe  are  sensibly  proportional  to  distances 
measured  along  the  horizontal  projection  of  the  pipe.  Hence  the  line 
of  virtual  slope  may  be  assumed  to  be  a  straight  line  without  error 
of  practical  importance. 

7.  Formulae  of  Darcy,  Hagen,  Thrupp,  Reynolds,  etc. — 

Darcy  arranged  the  results  of  his  experiments  in  a  table  drawn 
up  as  follows : 


Velocities  in  m./sec. 

•pw  •            A 

.10 

.12 

•T3 

.14 

to  3  m./sec. 

h 

}, 

h 

k 

D 

A 

L 

Q 

~L 

Q 

L 

Q 

L 

Q. 

140  PIPE  FORMUL/E. 

The  first  column  gives  the  several  diameters. 

The  second  column  gives  the  corresponding  sectional  areas. 

The  remaining  columns  give  the  several  velocities  of  flow 
from  4  ins.  (.  I  m.)  up  to  10  ft.  (3  m.)  per  second,  and  each 
velocity  column  is  subdivided  into  two  columns,  the  one  giving 

the  loss  of  head  {-=?}   per  unit  of  length,  and  the  other  giving 

the  discharge  (Q). 

An  examination  of  the  table  of  Darcy's  results  shows  that 
approximately  the  loss  //  is  directly  proportional  to  the  length 
L  of  pipe  under  consideration  and  to  the  square  of  the  velocity, 
z>,  and  is  inversely  proportional  to  the  diameter  d. 
Therefore 

L  4fL  v2  _v2 

<dv          d   -2:  "Lr> 


where  a  =       =       i  +  ,      (p.   127.) 

In  Hagen's  formula,  viz., 
h       av" 


the  values  of  a,  n,  and  x  vary  with  the  velocity,  the  diameter, 
and  with  the  roughness  of  the  surface.  The  results  obtained 
by  this  formula  are  in  accord  with  the  results  of  Pearsall's 
experiments  with  pipes  in  good  condition  and  of  diameters 
varying  from  .9  ft.  to  4  ft.,  when 

a  =  .0004,      n  =  1.87,      and     x  =  1.4, 

but  the  agreement  is  not  so  close  if  the  pipe  surface  is  very 
smooth. 

If  the  pipes  are  rough,  the  approximate  values  of  the  indices 
are 

a  —  .0007,      n  =  2,  and     x—  i.i, 


PIPE  FORMUL/E. 


141 


but  these  values   must   necessarily  vary  with    every  different 
class  of  pipe. 

Various  modifications  of  Hagen's  formula  have  been  pro- 
posed, and  perhaps  one  of  the  best  is  that  contained  in  a  paper 
by  Thrupp,  read  before  the  Society  of  Engineers  (London)  in 
1887.  It  may  be  written 

i  L        fm*\* 

—  =  cosec.  of  slope  angle  —  -r  =    — 

I  k  \  CV  I     ' 


2  —  m 
m 


being  substituted  for  x  when  m  is  small.      The 


values  of  n,  c,  x,  y,  and  z,   for  a  pipe  or  channel,   are  given 
by  the  following  table : 


Surface. 

n 

c 

X 

y 

z 

Wrought-iron  pipes  

1.  80 

0.004787 

0.65 

0.018 

0.07 

Riveted  sheet-iron  pipes.... 

1.825 

0.005674 

0.677 

New  cast-iron  pipes  -j 

1.85 
2.00 

0.005347 
0.006752 

0.67 
0.63 

Lc3.d  pines 

I   7< 

Orw>c  o*>/i 

f 

*•  /D 

.  uy  ^  ,£._-_$. 

O.O2 

Pure  cement  rendering  .  .*.  .  j 

1-74 
i-95 

0.004000 
0.006429 

0.67 

0.61 

Brickwork  (smooth)  

2.  GO 

O.OO774.6 

o  6  1 

(roii|r  h  )  

2  OO 

r\    f\CkRR/%  C. 

L»«  LMJOO^S 

0*02  5 

0.01224 

0.50 

Unplaned  plank  .... 

2  OO 

Q 

, 

o.  0004.5  1 

o.oi  5 

°-°3349 

0.50 

Small  gravel  in  cement.  .  .  . 

2.OO 

0.01181 

0.66 

0.03938 

0.60 

Large       "         "         "     

2.OO 

0.01415 

0.705 

0.07590 

I.OO 

Hammer-dressed  masonry. 

2.00 

0.01117 

0.66 

0.07825 

1.  00 

Earth  (no  vegetation)  

2.OO 

O   OT  C  lf\ 

OT> 

Rough  stony  earth  

2.00 

vJ»vJl  j^^J 

0.02144 

•1  * 
0.78 

Osborne  Reynolds  has  propounded  a  simple  law  of  resist- 
ance embracing  the  results  of  Poiseuille  and  Darcy,  and  taking 
into  account  the  effects  of  viscosity,  temperature,  etc.  This 
law  may  be  expressed  in  the  form  (the  units  being  a  foot  and 
a  second) 


i  =  the  slope  =      = 
L 


142  PIPE   FORMUL/E. 

in  which 

A  =  1.917  X  10,  B  —  36.8,  and  —  —  i  +  -O336/-J-  .000221^ 

/  being  the  temperature  in  degrees  centigrade.  Approxi- 
mately, the  index  n  is  I  if  the  critical  velocity  is  not  ex- 
ceeded, and  1.7  to  2  for  values  of  v  greater  than  the  critical 
velocity.  According  to  Unwin  the  index  of  d  is  not  exactly 
3  —  n  and  should  be  determined  independently.  For  a  rough 
surface  n  =  2,  for  a  smooth  cast-iron  pipe  n  =  1.9,  and  for  a 
lead  pipe  n  =  1.723 — a  limitation  which  is  analogous  to  that 
found  by  Froude  in  his  experiments  upon  surface  friction. 

It  may  be  noted  that  the  sum  of  the  exponents  of  v  and  d 
is  constant  and  equal  to  3. 

In  a  paper  read  before  the  Royal  Society  of  New  South 
Wales,  1897,  Knibbs  investigates  the  effects  of  temperature 
and  records  the  results  of  a  number  of  experiments,  but  the 
formula  he  deduces  is  too  complicated  to  be  of  much  practical 
value  and  requires  further  verification. 

Fournie  has  also  studied  temperature  effect  and  has  sug- 
gested a  formula,  but  his  results  are  not  complete  (Annales  des 
Pouts  et  Chaussees,  1898). 

Again,  a  simple  empirical  law  connecting  v,  m,  and  i 
may  be  expressed  in  the  form 


in  which  c  is  a  coefficient  whose  value  is  to  be  determined  by 

g  32.16  -513 

experiment.       Taking    c  =  —  _=  -^          "->=  —  7-.    then,    if 

wf       62.42  X/         / 

3//  =y,  this  formula  may  be  written 


z,  = 


For  values  of;/  from  .008  to  .018  the  results  are  practically 
the  same  as  those  obtained  by  substituting  the  same  values  for  n 
in  Kutter's  more  complicated  formula  (Chap.  Ill)  ;  but  while  the 


PIPE  FORMUL/E.  145 

two  formulae  closely  agree  in  ordinary  cases,  they  both  fail  in 
extreme  cases. 

The  formula  is  also  equally  applicable  to  open  channels 
(Chap.  Ill),  m  being  the  mean  hydraulic  depth ;  but  Tutton 
has  found  that  when  m  is  small,  and  especially  in  the  case  of 
open  channels,  it  is  preferable  to  use  the  modified  expression 

v  =  (Ii5_4  _  L \  ,/ ,-* 

\    n          ml 

Lampe's  well-known  formula  for  iron  pipes  is 

v  =  2O3.3i»W/f> 
while  Foss  gives  for  the  same  case 


In    1867,   M.  Levy  in  his   Theorie  d'nn   C  our  ant  Liquide^, 
the  units  being  a  metre  and  second,  gave  : 

for  new  cast-iron  pipes  v  =  3  6  .  4  j  ri(  i  +  Vr)  }  »  ; 

"   cast-iron  pipes  in  service    v  =  2O.${rt(i  -\-  3  Vr)\%. 
To  these  Vallot  added  in  1888: 
for  cleaned  cast-iron  pipes        v  =  32.5(^(1  -f-  Vr)\*. 

The  corresponding  formulae,  with  a  foot  and  seconds  units, 
are: 

v  —  9$.24\mt(i  +  .7809  Vm)\^; 
v  =  52.5i{w/(i  +  2.3427  Vm)\l\ 
+  .7809  Vw)|i 


Vallot  also  modified  the  expression  for  pipes  in  service,  and 
deduced 

v  =  64.7889*4  /^  in  metric  units, 
or 

v  =  96.  27  wa  2  ¥,  a  foot  and  second  being  the  units* 


144          GRAPHICAL  REPRESENTATION  OF  PIPE  FORMULA. 
Manning,  in  1890,  gave  the  formula 

1.486    4  i 

v  = m*i*, 

n 

n  being  the  same  as  in  Kutter's  formula,  Chap.  III. 
Flamant,  in  1892,  deduced  the  expression 

B   -| 

and  gave  the  following  values  for  c\ 

For  tin  pipe c  —  284. 5 

"    lead    " c  =  272.7 

"    glass  " r  =  262.  i 

"    wrought-iron  and  asphalted  pipe c  =  257.3 

"    new  cast-iron  and  tarred  pipe c  —  232.5 

"    lightly  incrusted  iron  pipes  in  service.,   c  =  205.4 

8.  Graphical  Representation  of  the  formula  v  =  cmji^. — 

The  preceding  formulae  are  special  applications  of  the  general 
expression 


in  which  the  coefficients  c,  x  and  y  for  any  series  of  experi- 
ments can  be  graphically  determined  in  the  following  manner: 
Taking  logarithms, 

log  v  =  log  c  +  x  log  m  +  y  log  i ; 

and  if  /\  is  a  particular  value  of  i  corresponding  to  a  value  ^, 
of  £', 

log  z>j  =  log  c  +  .r  log  w  -)-  ^  log  z'r 
Then 

log  ?'  —  log  v^—y  (log  z  —  log  /\), 

and  is  the  equation  to  a  straight  line,  the  rectangular  coordi- 
nates being  the  logarithms  of  v  and  of  i.  Selecting  any  set 
of  experiments  and  plotting  the  corresponding  values  of  log  v 


GRAPHICAL   REPRESENTATION  OF  PIPE  FORMULA.          145 

<md  log  2,  a  series  of  parallel  straight  lines,  inclined  at  a  con- 
stant angle  tan-1j^,  is  obtained.  For  all  the  velocities  corre- 
sponding to  log  i  =  O  or  i=  I,  i.e.,  at  the  intersections  of 
these  lines  with  the  axis  of  '  *  log  z»,  '  '  the  general  expression 
becomes 

v  =  cmx. 
Taking  logarithms  again, 

log  v  —  log  c  +  x  log  m, 

and  if  ?•,  is  the  value  of  v  corresponding  to  a  particular  value 
tn^  of  ;//, 

log  7^  =  log  c  +  x  log  mr 
Therefore 

log  ?'  —  log  ?/1  —  4r(log  m  —  log  m^ 


is  the  equation  to  a  straight  line,  the  rectangular  coordinates 
being  the  logarithms  of  v  and  of  m. 

Plotting  the  different  values  of  log  m  corresponding  to  the 
particular  values  of  log  v  in  question,  a  series  of  parallel 
straight  lines,  inclined  at  a  constant  angle  tan"1  x,  is  obtained. 
When  log  ;//  =  O,  or  m  =  I,  i.e.,  at  the  intersections  of  these 
lines  with  the  axis  of  '  '  log  v,  '  '  the  general  expression 
becomes 

v  =  c. 

Therefore 

log  v  =  log  c, 

and  the  coefficient  c  can  be  at  once  obtained  from  the  diagram, 
as  it  is  the  value  of  log  v  corresponding  to  m  —  I  and  i  =  I  . 
In  1896,  Tutton  completed  an  admirable  collaboration  of 
the  most  important  sets  of  experiments  on  pipe-flow,  more  than 
1000  in  number,  and  varying  widely  in  diameter  and  kind  of 
pipe. 


146 


PIPE-FLOW  DIAGRAMS. 


9.    Diagrams   Showing  Results    of    Experiments.  —  By^ 

means  of  the  method  just  described,  Tutton  has  plotted 
(Figs.  83-89),  representing  graphically  a  very  large  number 
of  experiments  on  pipe-flow  as  follows. 


v^LOG 


WOODEN  PIPES. 


V" 


UpGS.  JOF  m\  AND  {i 


.6      .8      3.       .2      .4       .0       .8      §1      .2      A       .6      .8   >  1.      .2      .4       .6      .8       0 
FIG.   83. — Flow  through  Wooden  Pipes. 

1.  Hamilton  Smith.     Series  10 m  =    .0263 

2.  Darcy  and  Bazin.         "       52*. m=    .303 

3.  Darcy  and  Bazin.         "        51 m  =     .505 

4.  Clarke,  Moon  Island  conduit m  =  1.50 

4  series,  22  experiments. 
Formula;  V  =  I29W66/-51. 


PIPE-FLOW  DIAGRAMS. 


.e 


QLAS 


PIPE 


LOGS  OF  TO  AND  i 


5.  Darcy. 

4.  Darcy. 

3.  Smith. 

2.  Smith. 


.8        1       .2       .4        .6        .8         1.       .2       .4        .6 
FIG.  84.  —  Flow  through  Glass  Pipes. 

Series  1  1  ........................    ;;/  =  .04075 


i.  Smith. 


7  

8  

.   m  —  .01  ccc 

Q.. 

.   m  =  .oiojx 

5  series,  32  experiments. 
Formula  :   v  =  141  to  169 


PIPE-FLOW  DIAGRAMS. 


NEW  WROUGHT  IRON  AND 
ASPHALT  COATED  PIPE. 


.6     .8      2.     .2      .4     .6      .8      1.     .2     w*   ^6      J3      0 
FIG.  85.  —  Flow  through  New  Wrought-iron  and  Asphalt-coated  Pipes. 


.4^.6     .8      3.     $     A 


f  Smith,  ) 

}  North  Bloomfield  f  " 
Iben,  Bonn  pipe  ....  ;//; 
|  Smith,  [ 

"/  North  Bloomfield  J   " 
Darcy.     Series  TO  ..  m: 
j  Smith,  /   m. 

I  North  Bloomfield  f 
Smith,  Texas  Creek,  m 

Lampe m 

Tubbs,  Rochester.  .  .  m~ 
Iben,  Sternchanze . .  m: 
Smith,  Humbug  pipe  ///: 
Smith,  I 

Cherokee  pipe  f 
Tubbs,  Rochester...  m. 

Gale in 

Rowland,  High  Heads  m- 

"  "          "        m: 

m. 


36  series,  195  experiments. 

Formula:  For  asphalt-coated  v  =  139  to  188  ;«-62/-55. 

For  new  wrought-iron    v  =  127  to  165  w-62t-s$. 


I. 

Darcy.     Series  i.  .  .  . 

tn  = 

.OI 

16. 

2, 

Smith.           "       6.  ... 

m=. 

.01307 

3- 

Smith.           "       5.... 

tu- 

.021325 

17- 

4- 

Darcy.           "       2.... 

rn— 

.02182 

18. 

5- 

Smith.           "      4.  •  •  • 

?n  = 

.0219 

6. 

Darcy            "       7  

in  — 

.02198 

19'. 

7- 

Smith.                   i,  2.. 

m  = 

.0219 

7- 

Smith.           "       3.  ... 

in  ~=^. 

.0218 

2O. 

j  Ehmann,      ) 

21. 

8. 

(  Hahnwald  j 

m  — 

.041 

22. 

9- 

Darcy.      Series  3.  ... 

m  = 

.0324 

22. 

10. 

(Crozet,                        ) 
I  Blue  Ridge  siphon  f 

m  — 

.0625 

23. 
24, 

i  i. 

Couplet              • 

,  m  — 

.  I  '*'?'> 

ii. 

Iben,  Deseniss  St.  .. 

i  J  3" 

.08375 

25- 

12. 

Darcy.     Series  8..  .  . 

m  = 

•06775 

26. 

13. 

Iben,  Schoen  St  

»i  = 

.12475 

27. 

Ij. 

Iben.     Series  5<*  ..'  .  . 

m  = 

.12475 

28. 

14. 

Ehmann,  Stuttgart.. 

»i  = 

.1655 

29. 

15- 

Darcy.     Series  9..  .  . 

m  = 

.16075 

30. 

1.22775 
=.251 

:.264 

:-23375 
=  •3075 

:-354 
=  •34325 


:-75 
•i.o 

:.O2O8 
:.O2O8 
:.0208 


PIPE-FLOW  DIAGRAMS. 


149 


TUBERCULATED  OR  RUSTED  PIPE  OF 
IRON  OR  LIGHT  MUD  DEPOSITS. 


1.2 


LOGS.  OF  m  AND  i 


.2      .4      .6      .8      3.     .2     .4      .6      .8     2.     .2     .4     .6      .8     1.     .2     .4     .6      .8     0 
FIG.  86.— Flow  through  Tuberculated  or  Rusted   Pipe  of  Iron  or  Light 

Mud  Deposits. 


1.  Iben,  Koppel  St.,  19  years  old.  .. m 

2.  Iben,  Schulweg,  19  years  old m 

3.  Couplet m 

4.  Darcy.     Series  12 m 

5.  Iben,  Schulweg,  1 3  years  old m 

6.  Fanning,  rusted  pipe m 

7.  Darcy.     Series  14. ..    m 

8.  Iben.  "      1 $a,  22  years m 

8.  Iben,  Strohhaus,  22  years m 

9.  Iben,  Carolinen,  15  years m 

10.  Darcy.     Series  19 m 

1 1 .  Couplet m 

12.  Iben,  Rotherbaum m 

12.   Ehmann ....  in 

12.  Iben,  Heidenkampsweg,  25  years m 

13.  Iben,  Hamm  St. . .   m 

14.  Iben,  Glacis  Chaussee. ;;/ 

14.  Duncan m 

15.  Bailey 


m  = 


•08375 

.1247 

.0888 

.02945 

.1247 

.020835 

.0652 

.2502-1- 

.2502 

.2502 

.1995 

.266 

.25 

.2075 -h 

.4167 

•25 
.250 
.25  + 
.4167 


PIPE-FLOW  DIAGRAMS. 


16.  Leslie m  = 

17.  Simpson.     Series  3 m  = 

18.  Leslie m  — 

19.  Couplet m  — 

20.  Simpson m  = 

21.  Greene m  — 

22.  McElroy,  Brooklyn  main m  = 

23.  Sherrerd,  Pequannock  main    . . .  , m  = 

23.  Sherrerd,  "  "      ;;/  = 

30  series,  132  experiments. 

Formula  :  For  light  tuberculations  v  —  87  to  132  r> 
For  heavy  tuberculations  v  =  31  to  80  w 


•3125 

•25 

•3333 

4 

•3957 
•75 
•75 
•75 
i.o 


>y 

// 

^ 

-.-%! 

.2 
•2. 
.8 
.8 
.4 

•2 

1. 
.5 
.0  ' 
.4 
.2 
0 
.8 
.6 

T.I 
) 

H 

^ 

/ 

X7 

^ 

^ 

/ 

^ 

$s 

•^ 

oV 

/^ 

/s 

/> 

h> 

* 

X 

y^ 

/ 

_^ 

>oV 

X" 

/ 

/^ 

/ 

// 

L^ 

k^1" 

0 

*x- 

/ 

^^ 

" 

> 

r.8 

^-^ 

^ 

^ 

^ 

§ 

^ 

^ 

*r 

? 

^ 

^ 

— 

^/ 

x^J 

FITZGERALD'S  EXPERIMENTS  ON 
THE  ROSEMARY  PIPE. 

XS 

^ 

^ 

^ 

Jo^' 

<^ 

>» 

^ 

-OGS 

.  Of 

mM 

iD  i 

r.      .2      .4      .6      .8      3.     .2      .4     .6      .8      2".      .2      .4     .6      .8      F.    '.2    \4      .6      .8      ( 
FIG.   87.  —  Fitzgerald's  Experiments  on  the  Rosemary  Pipe. 

1.  North  pipe.     Cleaned,  asphalted 

2.  Both  pipes.  Tuberculated 

4  series,  57  experiments. 
Formula  :  Cleaned,  asphalted,  v  — 
Tuberculated,  v  — 


m  —  i.oo 
m  =  i.oo 


PIPE-FLOW  DIAGRAMS. 


X 


NEW  CAST-IRON  AND  CEMENT 
LINED  PIPES. 


.6 
1.4 


AtfID  i 


•4.      .2     .4      .6      .8     3.     .2     .4      .G     .8     2.      3,     .4      .6     .8     1.     .2     ..4     .6     .8     0 
FIG.  88. — Flow  through  New  Cast-iron  and  Cement-lined  Pipes. 


2. 
3- 
4- 
5- 
6. 
6. 

7- 
8. 
8. 

9- 

10. 


Ehmann,  Neckar  St... m=  .0827  + 

Darcy.     Series  16 m  —  .067175 

Iben,  Wenden  St m=  .0835 

Iben,  Haller  St m=  .1245 

Darcy.     Series  17 m  —  .11237 

Darcy.         "       1 8 m  =  .  1 542 

Ehmann,  Stuttgart m  -  .20725  + 

Russell,  St.  Louis m  •=•  .25 

Darcy.     Series  22 m—  .4101 

Fanning,  cement-lined m=  .41674- 

Friend,  Seville m  —  -4375 

Woods,  Newton  (doubtful) m  —  .5 

Stearns,  Rosemary  pipe m  •=•  I .o 

13  series,  79  experiments. 
Formula  :  v  —  126  to  158  w66/-?1. 


PIPE-FLOW  DIAGRAMS. 


OLD,  NEW  AND  CLEANED 
CAST  IRON  PIPE. 


.8      0 


FIG.   89.  —  Flow  through  Old,  New,  and  Cleaned  Cast-iron  Pipes. 

1.  t)arcy.     Series  13  ____  ....................  ;;/  —  .02985 

2.  Meunier,  Torcy,  28-30  years  old  ............  ;//  =  .1107 

3.  Darcy.     Series  15  ...........  (.  .  .    ..........  m  =.  .0657 

4.  Meunier,  Nogent  sur  Seine,  new  ............  m  =  .1035 

5.  Coffin,  Taunton  main,  i\  years  old  ......  ,  ,  .  m  =  .625 

6.  Meunier,  Charenton,  2  years  old  ...........  m  =  .1640 

7.  Darcy.     Series  20   ...................  ....  m  =  .2007 

8.  Darcy.         "       21  ................  .........  m  ='   .2436 

9.  Forbes,  Brookline  main,  8  years  old  ........  m  =  -3333 

10.  Humblot,  i  series,  10  years  old  .............  in  =  .4921 

11.  Meunier,  Bercy  ..........................  m=  .4921 

12.  Humblot,  3  series,  6,  7,  and  12  years  old  .  .  .  m  —  .6562 

13.  Meunier,  Canal  de  1'Oise,  i  year  old  ......  m  =  .7382 

14.  Bruce,  Blane  Valley,  new  ...............  m  =  i.o 

16  series,  80  experiments. 
Formula:  v  =  96  to  148  w66/^1. 


TABLES   OF  PIPE  COEFFICIENTS.  15$ 

10.  Values  of  c,  x,  and  y  in  the  Formula  v  —  cma5i*/. 

Tutton    found    (see    Reynolds'    formula)   that,   in    the   general 
formula  v  —  cmxiy , 

x  -\-  y  =  a  constant  =  1.17, 
and  therefore 


The  values  of  c  and  y  he  has  tabulated  as  follows: 

c          y 

For  tin  pipe 1 89      .58 

For  lead  pipe 168      .58     Older  experiments  give  c  — 

189. 

For  brass,  zinc,  and  glass  pipe  165  .56  In  one  set  of  glass  experi- 
ments c  =  141. 

For  wrought-iron  pipe 160  .55  c  varies  from  127  to  165,  ap- 
proximating to  the  higher 
number. 

For  wood-stave  pipe 125      .51 

For    new   cast-iron   or  tarred 

pipe 130      .51     In  tarred  pipes  r  varies  from 

115  to  152,  the  values  be 
ing  about  the  same  as  in 
cast-iron  pipes  of  same 
size.  Benzinger  gives  for 
a  6o-in.  cast-iron  pipe  c 
=  129. 

For  pipe  in  service 104      .51     Generally  c  is  about  105.    In 

the    Rosemary    pipe    c  = 
117. 

For  tuberculated  pipe 30  to  80  .51 

For  lap-riveted  pipe 115      .51     c  varies  from   12510  135  for 

new  to  1 10  to  1 14  for  pipe 
in  service. 

For  rubber  and  leather  hose. .      160      .51 

For  wrought-iron  pipe  asphalt- 
coated 170      .55     In  some  cases  c  =  140,  and  in 

the  48-in.  pipe  c  =  199. 

For  large  brick  conduits 129      .52     Unobstructed  by  shafts. 

For  large  brick  conduits 91       .52     Fullerton  Avenue  conduit  of 

Chicago  water-supply. 

For  large  brick  conduits 1 10      .52     Chicago  land  tunnel. 


354  EXAMPLES. 

The  values  of  c  in  this  table  are  mean  values  and  neces- 
sarily vary  with  age  and  roughness. 

Throughout  the  analysis  of  these  experiments  the  total 
head  was  diminished  by  the  loss  of  head  at  entrance,  and  in 
the  cases  in  which  this  loss  had  not  been  found  by  means  of 

piezometers    it    has  been  calculated   from  —  • — ,  cc  being  the 

Cc  2<£" 

coefficient  of  contraction. 

Assuming  forj/  the  approximate  value  .5,  Tutton's  formula 
becomes 


Ex.  i.  The  head  over  the  sharp-edge  entrance  into  a  pipe,  1000  ft. 

long  and  passing  i  cu.  ft.  of  water  per  sec.,  is  9  ft.  Find  the  diameter, 
taking/  =  .0055. 

-^Ylj            4  x  .005$  x  loooN  _          49  /,           22 

~  64^2  "*                          //*          ~)  ~  16.  4' 


For  a  first  approximation,  disregarding  the  first  term  on  the  right- 
hand  side,  which  is  small  as  compared  with  the  second  term, 

49       22  _  49  j_ 
~    16.  121   d*  ~  88  d* 
and  d  =  .57319  ft. 

For  a  second  approximation 

_22_\ 


9  =  — - — = — —    1.5, 

I6.I2U/4  V  .573I9/ 

or  d*  = ^ x  39.8817, 

9.  16. 121 

and  d  —  .5787  ft. 

Ex.  2.  The  effective  height  of  the  grade  line  above  the  entrance  into 
a  clean  iron  3-in.  branch,  1000  ft.  long,  is  20  ft.  5  ins.  How  many  peo- 
ple will  the  branch  supply  with  20  gallons  of  water  per  head  per  day  of 
24  hours  ? 


/  ==  .005(1  +  —  L      ^  =  _L, 
>V         12  x  I)       150' 


EXAMPLES.  155 

• 
and  .  v  =  3i  ft   per  sec. 

• 

The  delivery  in  cu.  ft.  per  sec. 


-  7  4W         ~  64 
The  delivery  in  gallons  per  day 

_  II  x  6i  x  60  x  60  x  24  =  92,812^, 
64 

and  the  number  of  people  served  per  day  =  —  -    '?  =  4640!, 


20 


or  4640. 


Ex.  3.  Find  the  proper  diameter  of  a  rough  pipe  to  give  60,000,000  of 
gallons  every  24  hours^the  slope  of  the  pipe  being  i  in  800. 

22  d~  60,000,000 


Ir- 


or 


6J  .  60  .  60 .  24' 
14000 


99 


h        avn          i 

Using  Hagen  s  formula,  viz.,  —  =  —    —  =  -  —  , 

L        dx         800 

and  taking  a  —  .0007,     «  =  2,     and     .r  =  i.i, 

JL    _'-00°7    a        .0007  /  14000  y 
~  ; 


</*,  =  800  x   . 

Therefore  ^  =  6.22  ft. 

Ex.  4.  What  should   be  the  slope  of  a  24-in  wooden-stave  pipe  to 
give  5,940,000  gallons  per  day  ? 

22  (2)2  5,940,000 

7     4    '     ~  6J  .  24  .  60  .  60  ~ 

,ind  v  =  3|  ft.  per  sec. 

Take  the  formula  v  =  cm^—fty. 

By  the  Table,  c  =  125     and    y  —  .51. 

/2\.66   .51 

Therefore  $\  —  I25(-J    f    , 

and  i  =  .002212,  or  about  22  in  10,000 


IS6  TRANSMISSION  OF  ENERGY. 

• 

ii.  Transmission  of  Energy  by  Hydraulic   Pressure. — . 

Let  Q  cu.  ft.  of  water  per  second  be  driven  through  a  pipe  of 
diameter  d  ft.   and   length  L  ft.  under  a  total  head  of  H  ft. 
Also  let  n  per  cent  of  the  total  head  be  absorbed  in  overcoming 
the  frictional  resistance  in  the  pipe.      Then 

the  head  expended  in  useful  work  =  H  —  h 

=  4---). 

\  100'* 

and  the  efficiency  =  — 7^ —  =  i 


H  100 

Again, 

nH        h        4fL  ^  ^fLff 
100   "  d    2g         nW 

79 

Since  Q  =  --  z/,  and  g  is  assumed  to  be  32, 


_ 

10          fL 

and  the  total  available  work  in  foot-pounds  per  second 


If  TV^  is  the  number  of  horse-power  delivered  at  the  end  of 
the  pipe, 


wQH/  i  \          /          n\       nH3d5 

550 


w  ji  \      ^/ 

550  A1  "ioo/~28\I  " 


an  equation  giving  the  distance  L  to  which  N  horse-power  can 
be  transmitted  with  a  loss  of  n  per  cent  of  the  total  head. 
Again, 

~  .  h  2fL  v2  2fLw  v2 

tke  efficiency  =  I  —  =.  =  i  --  ^r        =  i  —  -- 

H  gE    d  fj     pd* 


TRANSMISSION  OF  ENERGY.  157 

p  (=  wH)  being  the  pressure  corresponding  to  the  head  H. 
The  efficiency  diminishes  as  v  increases  and  therefore,  so  far 
as  efficiency  is  concerned,  it  is  advantageous  to  transmit  energy 

v2 
at  a  low  speed.      Again,  the  efficiency  is  constant  if  —  ^  is  con- 

stant. 

Assuming  this  to  be  the  case,  take  v2  =  c2  .  pd.      Then  the 


total  energy  transmitted  =  wQH  '  —  w  —  —vH 

4 


If  it  be  also  assumed  that  the  thickness  /  of  the  pipe-metal 
is  so  small  that  the  formula 

pd  =  2ft 

holds  true,  f  being  the  circumferential  stress  induced  in  the 
metal,  then 

the  energy  transmitted  =  -—  p%d% 

4 

ncf'td    . 
=  -~-    «P<* 

cf'V    .— 
=  -i-"l* 

V  being  the  volume  of  the  pipe  per  unit  of  length. 

Hence,  for  a  given  volume  V  of  metal  and  a  constant 
efficiency,  the  energy  transmitted  is  a  maximum  when  pd  is  a 
maximum. 

If  /  is  increased  beyond  a  certain  limit,  the  ratio  -=  is  no 

longer  small  and  the  thickness  /  will  have  a  greater  value  than 
that  given  by  the  equation  pd  =  2f't.  Then  the  cost  of  the 
pipe  will  also  increase.  On  the  other  hand,  if  d  is  increased, 


158  TRANSMISSION  OF  ENERGY. 

the  ratio   -^,   and   therefore   also  the  pressure  /,   will   remain 

small,  and  thus  the  cost  of  the  pipe  will  not  increase.  Hence 
it  is  more  economical  to  employ  large  pipes  and  low  pressures 
than  small  pipes  and  high  pressures. 

The  demand  for  hydraulic  power  in  large  cities  has  led  to 
the  laying  down  of  networks  of  mains  through  which  water  is 
conveyed  under  pressure  and  is  distributed  to  the  consumer 
for  various  industrial  purposes.  Since  the  loss  of  head  due 
to  frictional  resistance  is  approximately  proportional  to  the 
square  of  the  velocity,  and  since  also  the  momentum  of  the 
moving  fluid  must  not  be  so  great  as  to  make  excessive  shocks 
possible,  high  velocities  cannot  be  allowed  in  the  mains  or  in 
the  machines  operated  by  the  pressure-water  except  for  very 
short  distances.  Thus,  the  velocity  of  flow  in  the  mains  is 
limited  to  6  ft.  per  second,  and  rarely  exceeds  8  ft.  per 
second  in  the  machines.  In  London  the  average  rate  is  4  ft. 
per  second.  Again,  the  quantity  of  power  conveyed  by  a 
single  main  cannot  be  great.  Hence  the  hydraulic  distribu- 
tion of  power,  in  which  the  pressure  of  water  is  directly  utilized, 
is  especially  adapted  for  machines  with  slow-moving  rams, 
which  are  intermittent  in  action  and  which  work  only  for  short 
intervals  of  time,  as,  for  exmple,  in  lifting  and  pressing  opera- 
tions and  when  a  great  effort  is  to  be  exerted  through  a 
short  distance.  In  London  the  pressure  in  the  mains  is  750 
Ibs.  per  sq.  in.,  but  in  the  more  recent  distributions  in  Man- 
chester and  Glasgow  the  pressure  is  uoo  Ibs.  per  sq.  in.  The 
working  stress  in  the  cast-iron  mains,  the  largest  in  use  being 
7£  ins.  in  diameter,  is  2800  Ibs.  per  sq.  in.,  and  they  are 
generally  tested  to  2500  Ibs.  per  sq.  in.  before  laying  and  to 
about  1000  Ibs.  per  sq.  in.  after  laying.  The  thickness  t  in 
inches  of  cast-iron  main  of  d  ins.  diameter  under  a  water- 
pressure  of  /  Ibs.  per  sq.  in.  may  be  determined  by  the 
formula 

t  —  .QOOjSpd  -f-  .25  in. 


EXAMPLES.  159 

Another  formula  gives  /  =  .0024^  -f-  .75  in.,  /  being  the 
pressure  in  atmospheres. 

With  suitable  joints,  and  drawn  tubes  of  steel  with  a  tenacity 
of  15,000  Ibs.  per  sq.  in.,  the  hydraulic  system  of  distribution 
could  be  greatly  extended. 

Again,  for  an  hydraulic  pipe  or  press 

f—      HinrT~2 I         ^3 


where  />0  ,  /j  are  the  intensities  of  pressure   at  the  outer  and 

inner  surfaces; 

f  is  the  intensity  of  stress  at  the  radius  r  ; 
rQ  ,  rl  are  the  radii  of  the  outer  and  inner  surfaces. 
(See  Appendix,  Bovey's  "Theory  of  Structures.  ") 

Ex.  i.  An  accumulator  supplies  a  pressure  of  700  Ibs.  per  sq.  in. 
What  length  of  8-in.  pipe  will  deliver  200  H.P.  of  useful  energy  with 
a  loss  of  20  per  cent  ? 

250  H.P.  enter  the  pipe.  Therefore,  if  Q  is  the  delivery  in  cu.  ft.  of 
water  per  sec., 


=<=-. 
252  7     4\3 


and  V  =  ~-    ft.  per  sec. 


Take       /  =  .001;    i  +  I  =    /     ,  for  a  clean  iron  pipe. 

\  12     X     •§-/  rAi-vrv 

Then 

x  250  =  loss  =  62!  .  — -  . 

ioo  252  f  04      550 

L  being  the  length  of  the  pipe. 

Therefore  L  —  78,293.7  ft.  =  14.8  miles. 

Ex.  2.  The^ efficiency  of  an  engine  is  .6;   it  burns  2  Ibs.  of  coal  per 
hour  per  H.'P.^'and  works  16  hours  a  day  for  300  days  in  the  year.     The 


160  PRESSURE  DUE    TO  SHOCK. 

cost  of  the  engine  is  $12  per  H.P.,  and  the  cost  of  the  coal  $3  per  ton. 
An  amount  of  4500  gallons  of  water  per  minute  is  to  be  raised  a  vertical 
height  of  200  ft.  What  must  be  the  minimum  diam.,  D,  of  the  pipe, 
assuming  that  the  cost  of  the  piping  is  §D  per  lineal  foot,  and  that 
/  =  .0064  ? 

Let  h  feet  be  the  frictional  loss  of  head. 

41500  22  Z)2 

rhen'smce          =-~v=12' 


_  4  x   .0064  x  200  i_  (YT8-)  _  2  /i68\2    i 

D        ~  64  "  ~7J*  ~  2^  \ir)  'W' 

Again,  let  N  be  the  number  of  H.P. 

Then  N  =  -  *2  '    ^(200  +  //) 

:>       55° 

25  j  2  /r68\2    i 

=  -     ]   200  +   —     --         y,r 

n(  25  \.  ii  I   D* 

Cost  of  coal  capitalized  at  5^  =  N.  2    '  ,,^°°  -'  ~  •  ~-°  =  $288^". 

Cost  of  engine  =  f>i2Ar. 

Cost  of  piping  =  $20o/>. 

Total  prime  cost  =  3ooAT  +  2ooZ^ 

(  A68V    i    ) 

=  300    X    f|  |   200  +   {—]     W  [   +  200A 

which  must  be  a  minimum. 

Therefore  O  —  —  300  x  f  \  f  —  -J    —  +  200, 

and  D  =  3.98  ft. 

Hence,  a!so,  k  =         -*  =  .0,868  ft. 


No.  of  H.P.  =  N  =  454  .  59. 
Capital  cost  =  $137,173. 

12.  Pressure  Due  to  Shock.  —  Water  flows  through  a  line 
of  piping  with  a  velocity  of  v  ft.  per  second,  and  at  a  certain 
point  the  motion  is  suddenly  arrested  by  the  closing  of  a  valve, 
developing  a  sudden  increase  in  the  pressure  at  the  valve  of 


PRESSURE  DUE   TO  SHOCK.  161 

f  Ibs.  per  sq.  in.  Water  being  slightly  compressible — losing 
TV*°f  its  bulk  under  a  pressure  of  2  tons  (of  2240  Ibs.)  per 
square  inch — a  compression-wave  starts  from  the  valve  and 
moves  backwards  throughout  the  whole  length  L  ft.  of  the 
moving  column  of  water.  The  water  still  enters  the  pipe  for 
the  period  of  /  seconds,  during  which  the  compression  con- 
tinues. 

Let  a  ft.  be  the  sectional  area  of  the  water-column ; 
"     x  ft.  be  the  diminution   in  the  length  L  of  the  wafer 

column ; 
' '     K  be  the  modulus  of  cubic  elasticity  of  water  =  300,000 

Ibs.  per  sq.  in. 
Then 

*    L 

L~~  1C 


x  —  vt 


w 
and        1440  ft.   =  momentum  of  the  fluid  mass  =  —  aLv. 

o 

Hence 

i    w  L  i    w  L  i     w  L2  f 

~  144  J  72'    ~  144  £-  ~P*    ~  144  ^~ "?  J?' 

and 

-  =  velocity  of  the  wave-propagation  =  /*/€__. 
t  \     w 

Substituting  the  values  of  g,  K,  and  w,  the  velocity  of  wave- 
propagation  is  found  to  be  about  4/20  ft.  per  second,  which  is 
also  the  velocity  of  sound  in  water. 

Ex.  A  volume  of  water  50  ft.  in  length,  flowing  through  a  pipe  with 
a  velocity  of  24  ft.  per  sec.,  is  quickly  and  uniformly  stopped  in  one  tenth 
of  a  second  by  closing  a  stop-valve.  Find  the  increase  of  pressure  per 
sq.  in.  in  the  pipe  near  the  valve. 

The  pres.  per  sq.  in.  =  — —  .  -   —  .  *          =  162.76  Ibs. 


32        144 


1 62  FLOW  IN  PIPE   CONNECTING    TWO  RESERVOIRS. 

13.  Flow  in  a  Pipe  of  Uniform  Section  and  of  Length  Z, 
connecting  two  Reservoirs  at  Different  Levels.  — Let  z  ft.  be 

the  difference  of  level  between  the  water-surface  in  the  two 


reservoirs. 


FIG.  90. 

The  work  done  per  second  is  evidently  equal  to  the  work 
done  by  the  fall  of  wQ  Ibs.  of  water  through  the  vertical  dis- 
tance z,  and  is  expended  — 

(1)  In  producing  the  velocity  of  flow  v  ft.  per  second,  which 

requires  a  head  of  z^  ft.  and  an  expenditure  of  wQzl 
ft.  -Ibs.  of  work  per  second; 

(2)  In  overcoming  the  resistance  at  the  entrance  from  the 

upper  reservoir  into  the  pipe,  which  requires  a  head 
of  -s-g  ft.  and  an  expenditure  of  wQzz  ft.  -Ibs.  of  work 
per  second; 

(3)  In  overcoming  the  fractional  resistance,  which  requires 

a  head  of  zz  ft.  and  an  expenditure  of  wQz^  ft.  -Ibs. 
of  work  per  second.  Thus 

wQz  —  wQzl  +  wQz2  +  wQz^  , 
or 


Now  zl  —  —  ft.,    and  the  corresponding  energy  wQzl  is 

ultimately  wasted  in  producing  eddy  motions,  etc.,  in  the  lower 
reservoir. 

v* 
#2  may  be  expressed  in  the  form  n  —  ft.  ,  n  being  a  coeffi- 


CHEZY'S  LONG-PIPE  FORMULA. 


cient  whose  value  varies  with  the  nature  of  the  construction  of 
the'entrance  into  the  pipe.  If  the  pipe-entrance  is  bell-mouth 
in  form,  n  =  .01  or  .02,  but  if  it  is  cylindrical,  n  =  .49. 
Finally, 


_ft 

—  --  it.  —  —r        it.  , 

m    iv  d   2g 


takine  —  —  =  f—  »  as  is  usual  in  practice.      Hence 

W  2 


T 

since  Q  = V,  and/g'is  assumed  to  be  32. 

For  given  values  of  Q  and  z  a  first  approximate  value  of  d 
may  be  obtained  from  the  last  equation  by  neglecting  the  term 

_|_  //).      Call  this  value  dv ,  and  substitute  it  for  the  d 


in  the  term  —  7-  within  the  brackets.     A  second  approximation 
d 

may  now  be  made  by  deducing  d  from  the  formula 


and  the  operation  may  be  again  repeated  if  desired. 

Generally  speaking,    I  +  n  is  usually  very  small  as  com- 

pared with  ~  ,  ,  and  maybe  disregarded  without  error  of  prac- 

tical importance. 

The  formula  then  becomes 

4fL  v2 
d   2g' 

which  is  known  as  Chezy's  formula  for  long  pipes. 


1  64  LOSSES   OF  HEAD. 

The  term    I  -j-  ;/  need   only  be  taken  into  account  in  the 
case  of  short  pipes  and  high  velocities. 

Ex.  The  difference  of  level  between  the  water-surfaces  of  two  reser- 
voirs, connected  by  a  24-in.  pipe  6J  miles  in  length,  is  172^  ft.  The 
pipe,  having  been  in  use  for  some  time,  has  its  inside  surface  coated 
with  a  deposit,  and  no  special  provision  is  made  to  diminish  the  resist- 
ance at  the  upper  end.  Determine  the  discharge  into  the  lower  reser- 
voir in  gallons  per  hour. 


/  =  .01  ( 


Take  =  .01    i  + 

Then       .7* 


and  v  =  4  ft.  per  sec. 

Therefore  the  discharge  in  cu.  ft.  per  hour 


and  the  discharge  in  gallons  per  hour 

=  45.2571-  x  6i  =  282,857}. 

14.  Losses  of  Head  due  to  Abrupt  Changes  of  Section, 
Elbows,  Valves,  etc.  —  When  the  velocity,  or  the  direction  of 
motion  of  a  mass  of  water  flowing  through  a  pipe,  is  abruptly 
changed,  the  water  is  broken  up  into  eddies  or  irregular 
motions  which  are  soon  destroyed  by  viscosity,  the  correspondr 
ing  energy  being  wasted. 

CASE  I.  Loss  due  to  a  sudden  contraction.  (Art.  17, 
Chap.  I.) 


FIG.  91.  FIG.  02. 

(a]  Let  water  flow  from  a  pipe  (Fig.  91),  or  from  a  reser- 
voir (Fig.  92)  into  a  pipe  of  sectional  area  A. 


LOSSES  DUE   TO  SUDDEN  CONTRACTION,  ETC.  165 

Let  cc  be  the  coefficient  of  contraction. 
*  Then  the  area  of  the  contracted  section  —  ccA ,  and 


I   Iv 

the  loss  of  head  =  —  I  --  v  } 
2\c          I 


2gcc 


•v 

where  m  =  I—  — 


The  value  of  m  has  not  been  determined  with  any  great 
degree  of  accuracy;  but  if  cc  =  .64,  then  m  —  .316.  The 
value  of  cc  is  sometimes  obtained  from  the  formula 


2.618- 


When  the  water  enters  a  cylindrical    (not    bell-mouthed) 
pipe  from  a  large  reservoir,  the  value  of 
_,  m  is  about  .  5°5- 

(b)  Let    the    water    flow    across    the 
abrupt  charfge  of  section  through  a  central 
FIG.  93.  orifice  in  a  diaphragm  placed  as  in  Fig.  93. 

Let-0  be  the  area  of  the  orifice. 
Then  c<a  is  the  area  of  the  contracted  section,  and 


2 
the  loss  of  head 


I  A          \2  z>2  v> 

=  =  (—  -  i)  —  =  m—  , 

(A          \2 

where  m  —  I—  —  I  J  . 
/ 


1 66  LOSSES  DUE   TO  SUDDEN  CONTRACTION,  ETC. 

According  to  Weisbach, 


A  ~ 

.  i 

.  2 

-3 

•4 

•5 

cc  = 

.616 

.614 

.612 

.610 

.607 

m  = 

23I-7 

50.99 

19.78 

9.612 

5.256 

a 

.6 

•  7 

.8 

.  -9 

1.  00 

cc  = 

.605 

.603 

.601 

.598 

.596 

m  = 

3-077 

1.876 

1.169 

•734 

.48 

(V)  A 

diaphragm   with 

a   central  r 

orifice  of 

area  a, 

olaced  in  a 

cylindri- 

\ 

cal    pipe 

of   sectional    area 

J 

A   as  in 

^jjjfif^^10  ^ 

Fig.  94. 

FIG. 

94- 

The  *  *  contracted  area  ' '  of  the  water  =  cca  and 

i    IvA         V       ^  I A          V 

the  loss  of  head  =  —  I v  \   =•  —  I 1 1 

2g\cca          1        2g\cfa         I 


v* 


I A          \2 
where  /«  =  (^ ij  . 


Generally  m  must  be  determined  by  experiment,  but  Weis 
bach  gives  the  following  results : 


•5.- 

.1 

.2 

•3 

•4 

•5 

m  = 

.624 
225.9 

.632 

47-77 

•643 
30.83 

.659 
7.801 

.681 
3-753 

*,=    -712 

*0  =     1.796 


.7 

-755 
•  797 


.8 

.813 
.29 


•9 

.892 
.06 


i. oo 

1. 00 

oo 


LOSSES  DUE    TO  ABRUPT  ENLARGEMENT,  ELBOWS,  ETC.     167 
CASE  II.    Loss  due  to  a  Sudden  Enlargement       (Fig.  95.) 

Let  A^  =  external  area  of  small  pipe. 
"  A2  =  .         "        "     "  large     4< 


z  /A 
loss  of  head  = 


I    /2         \         v    /2         \ 
=  -  (-^  -  v)    :=  -  (-  -  ij 


where  m  =  l^r-  — 


NOTE.  —  The  losses  of  head  in  Case  I  (a)  and  in  Case  II  may  be 
avoided  by  substituting  a  gradual  and  regular  change  of  section  for  the 
abrupt  changes. 

CASE  III.  Loss  of  Head  due  to  Elbows.  (Fig.  96.)  —  The 
loss  of  head  due  to  the  disturbance  caused  by  an  elbow  is  ex- 
pressed by  Weisbach  in  the  form 


m 


0  0 

where  m  =  .9457  sin2  —  -f  2.047  sin4  ~> 

0  being  the  elbow  angle. 

Weisbach  deduced  this  formula  from  the  results  of  experi- 
ments with  pipes  1.2  in.  in  diameter. 

The  velocity  v^  with  which  the  water  flows  along  the  length 
AB  may  be  resolved  into  a  component  v  with  which  the  water 
flows  along  BC  and  a  component  u  at  right  angles  to  the 
direction  of  v.  The  component  u  and  therefore  the  corre- 

u2 
spending  head,   viz.,    -  —  ,  is  wasted.      The  component  u  evi- 

dently diminishes  with  the  angle  0  and  becomes  nil  when  a 


1 68 


LOSSES  DUE   TO  ELBOWS,  BENDS,  ETC. 


gradually  and  continuously  curved  bend  is  substituted  for  the 
elbow. 


CASE  IV.  Weisbach  gives  the  following  empirical  formula 
for  the  loss  of  head  at  a  bend  in  a  pipe,  0  being  the  angle  of 
curvature : 


h  =,  —  * 


where 


=  .131  + 


(  d\\ 

"1.  847(^1 


for  a  circular  pipe  of  diameter  d,  p  being  the 
radius  of  curvature  of  the  bend,  and 


FIG.  97. 


for  a  pipe  of  rectangular  section,  $•  being  the  length  of  a  side 
of  the  section  parallel  to  the  radius  of  curvature  (p)  of  the  bend. 
According  to  Navier, 

^  =  (.0128 +  .0186^—, 

<$ 

R  being  the  radius  and  L  the  length  of  the  bend  measured 
along  the  axis. 

As  a  result  of  recent  experiments  by  Gardner  S.  Williams 
and  others  (Proc.  Am.  Soc.  C.  E.,  May,  1901)  it  is  claimed 
that,  down  to  a  limit  of  2j  diameters,  curves  of  short  radius 
offer  less  resistance  to  flow  than  do  curves  of  longer  radius, 
which  is  contrary  to  the  ordinary  hypothesis. 


LOSSES  DUE   TO  SLUICES,  VALVES,  ETC. 


169 


CASE  V.  Valves,  Cocks,  Sluices,  etc. — The  loss  of  head  in 
each  of  the  cases  represented  by  the  several  figures  may  be 
traced  to  a  contraction  of  the  stream  similar  to  the  contraction 

which  occurs  in  the  case  of  an  abrupt  change  of  section.      The 

2,2 
loss   may  be  expressed  in   the  form  m —  ,  and  the  following 

o 

tables  give  the  results  obtained  by  Weisbach : 

(a)  Sluice  in  Pipe  of  Rectangular    Section.      (Fig.   98.) 
Area  of  pipe  =  a;  area  of  sluice  —  s. 

.9     .8      .7        .6        .5       .4       .3       .2       .1 

m  —  .00  .09  .39  .95   2.08  4.02  8.12  17.8  44.5  193 
FIG.  98. 

(ft)   Sluice  in   Cylindrical  Pipe.      (Fig.    99.) 
s  —  ratio  of  height  of  opening  to  diameter  of  pipe. 

s=     i      .875     .75    .625       .5       .375       .25        .125 
m  =  .00     .07      .26      .81      2.06    5.52     17.00     97.8 

(c)   Cock  in  Cylindrical  Pipe  (Fig.  100). 
s  =  ratio  of  cross-sections; 
6  =  angle  through  which  cock  is  turned. 


FIG.  99. 


FIG.  101. 

15°       20°      25°       30°       35° 
.772     .692     .613     .535     .458 


75      1.56 


45 
315 


50° 

•25 


55° 
.19 


3.1       5-47     9-68 

60°       65°       82° 
.137     .091       oo 


31.2     52.6       106      206      486 


CO 


LOSSES  DUE   TO   CHANGES   OF  SECTION,  ETC. 


(d)    Throttle-valve  in  Cylindrical  Pipe  (Fig.   101). 

6  =  angle  through  which  valve  is  turned. 
.  If0=   5°      10°     15°      20°      25°       30°       35°      40° 
m  —  .24     .52     .90     1.54     2.51      3.91      6.22      10.8 

If0=    45°       50°       55°      60°     65°     70°     90° 
m=iS.?     32.6     58.8     118     256     751      oo 

CASE  VI.  The  fall  of  free  surface-level,  or  loss  of  head,  due 
to  sudden  changes  of  section,  frictional  resistance,  etc.,  may- 
be graphically  represented  as  in  Fig.  102. 

-*• 


FIG.  102. 

Let  a  length  of  piping  AE  connect  two  reservoirs,  and  let 
//  be  the  difference  of  surface-level  of  the  water  in  the  reser- 
voirs. 

Let  Ll  ,  rl  be  length  and  radius  of  portion  AB  of  pipe. 
"  L2,  r2   "        "         "        "       "        "        BC  "      " 


''  Z4,  r4   "        "         "        "       "        "       DE  "      " 
<4  ul  ,  «?,  «3,  «4  be  the  velocities  of  flow  in  AB,  BC,  CD, 
DE,  respectively. 


LOSSES  DUE   TO   CHANGES  OF  SECTION,  ETC.  171 

The  reservoir  opens  abruptly  into  the  pipe  at  A . 
'*    There  is  an  abrupt  change  at  B  from  a  pipe  of  radius  rl  to 
one  of  radius  r2. 

There  is  an  abrupt  change  at  C  from  a  pipe  of  radius  r.2  to 
one  of  radius  rv 

o 

At  D  the  water  flows  through  an  orifice  of  area  A  in  a 
diaphragm.  At  E  the  velocity  of  the  water  as  it  enters  the 
lower  reservoir  is  immediately  dissipated  in  eddies  or  vortices. 

Draw  the  horizontal  plane  amnop  at  a  distance  from  the 
water-surface  in  the  upper  reservoir  equal  to  the  head  due  to 
atmospheric  pressure. 

Draw  vertical  lines  at  A ,  B,  C,  D,  E.      Take 

u  a 
ab  =loss  of  head  at  the  entrance  A  =  .49—  ; 

qc  =   "     "       "     due  to  friction  from  A  to  B    —2^1^-Ll ; 

' 


cd=   "     "       "     due  to  change  of  section  at  B—(T~—  I)  — : 

W      /.V 

r*  =   "     "       "     due  to  friction  from  B  to  £7    =2lU-*-L^ 


ef=   "     "       "     due  to  change  of  section  at  £7=. 316—  ; 

2S 

sg=   u     «       a     due  to  friction  from  C  to  D    =^~  .  U^-L- 

r*    ig 

gh=   "     "       "     due  to  change  of  section  at  D=  ( — 7  —  1)  — 

\ceA       I  2g 


tk  =   "     "       "     due  to  friction  from  D  to  £    =^--±L4 ; 

u  2 
^/=   "     "       ««     corresponding  to  w  =»— . 


I72  LOSSES  DUE   TO  CHANGES  OF  SECTION,  ETC. 

Through  /  draw  a  horizontal  plane  Ix.  This  plane  must 
evidently  be  at  a  distance  from  the  water-surface  in  the  lower 
reservoir  equal  to  the  pressure-head  due  to  the  atmosphere. 

Then  the  total  loss  of  head  =  lp 

=  ab  +  cd+ef  +  gh  +  kl  +  qc  +  re  +  sg+tk, 


.  ,  t 

,   2g        r  r^   2g        r  r3  2g  r,  2g 


-49  ,/^        \'J    ,  ^6  ,  /^!_  .\J1   ,    i  I 
r7+  U?~'    J  I-;*"*'   rs4  'hl^  '     V  ^'  +  ^7 


The  broken  line  abcdefghkl  is  the  hydraulic  gradient. 

Ex.  A  clean  6-in.  pipe,  ^400  ft.  long,  containing  a  60°  bend  with  a 
i2-in.  radius,  a  90°  bend  with  a  ;2-in.  radius,  and  a  120°  bend  with  a 
48-in.  radius,  discharges  i  cu.  ft.  of  water  per  sec.  into  a  clean  12-in.  pipe, 
200  ft.  long,  which  again  discharges  into  a  clean  4-in.  pipe,  500  ft  long, 
containing  four  sharp  knees,  viz.,  one  of  60°,  one  of  90°,  one  of  120°,  and 
one  of  150°.  Find  the  total  head  wasted  at  the  pipe  entrance,  at  the 
bends,  knees,  sudden  changes  of  section,  and  in  the  straight  lengths. 

Let  s/i,  772,  2>3  be  the  velocities  of  flow  in  the  first,  second,  and  third 
lengths,  respectively.  Then 

22    I    /I  \f  22     I  22    I  /I  \2 

--     -  }   Vi    =    I    =    -      —  (I2)772  =  —  -    --    ]V» 

7  4  \2/  7   4V    '  7   4\3/ 


EXAMPLES.  173 

and 

56  ,  14  ,  126  , 

*  v\  =  —  ft.  per  sec.,     V*  =  —  ft.  per  sec.,     7/3  = ft.  per  sec. 

ii  ii  ii 

Head  wasted  at  pipe  entrance  =  -  f  — )  —  —  .20332  ft. 

0        V* 

The  head  wasted  at  a  bend         =  mb—^-^  — , 

1 80  ig 


where  mb  =  .131  +  1.847^ — J*. 


2/J            26          4  ' 

d          6           i 

2/J  ~~    144    ~"   24' 

mb  =  .14544  ; 
mb  —  .13102727; 

d        6        i 
zp   ~96  ~  6' 

1)lb   —    .131113. 

Hence 

head  wasted  at  60°  bend  =  .14544     x  3\°o   x  ^  x  (ff)2  —  .019632  ft.,. 
90°      "     =  .130273  XyVir    x  -fa  x  (H)a  =  - 0265303  ft., 
120°      "      =  .131113  x  jfS  x  ,V  x  (ff)2  =  .035396  ft., 
and  the  head  wasted  in  bends  —  .081558  ft. 

The  head  wasted  at  a  knee  =  nik — , 

where  mk  —  -9457  sin2  --  +  2.047  sin*  — . 

For  a    60°  knee    (p  =  120°,      mk  =  1.8607 

90°     "       0  —    90°,     nik  =     .9846 

"       1 20°     " ...  0  =    60°,     nik  =     .36436 

1 50°     "       0  =     30°,     mk  =     .07254 

Then 

head  wasted  at    60°  knee  =  1.8607     x  ^(W)a.=  3-8l463  ft., 

"          "          90°    "     =    .9846    x  ^V  O^6)2  =  2.01853'" 
"  "  I2°°     "      =    .36436  x  ^VGW)2  ==  -74^97  " 

I500       "         =       .07254    X    TsH-TT6)2   =   -14871    " 

and  the  head  wasted  in  knees  =  6.72884  ft. 
Head  wasted  at  junction 

between    6-in.    and    i2-in.  pipes  —  ^V  (ff  —  if)2  =  .22778  ft., 

"          12-in.    and      4-in.  pipes  =   '—-( ^J       =  .64783  ft.» 

64  \  1 1  / 

and  the  head  wasted  at  sudden  changes  of  section        =  .87561  ft. 


274  NOZZLES. 

For  straight  lengths 

take/  =  .005(1  +  7^r^  =  '-^i  for  6-in.  pipe, 

12-in.     " 


/  i      \       .065 

/  =  .005    i  H — — -     " 

°\  12    X    l)  12 


Then  head  wasted 


•°35 
4  x  -fUoo    i 

in  ist  length  =-  -in 1 77      =  7-5592  ft., 


.065 
4  x  ^200        .     .  „ 

12  I    /H\2  . 

2d      "  -z-l^      =7.01929  ft., 

i  64\ny 


.025 

4  x       ; 


and  the  frictional  loss  of  head  =  91.14329  ft. 
Hence  the  total  head  wasted 

=  .081558  -f  6.72884  +  .87561  -f  91.45729  =  99.1433  ft. 

16.  Nozzles.  —  Let  a  pipe  ABy  of  length  /  and  diameter  d, 
lead  from  a  reservoir  h  ft.  above  the  end  B,  Fig.  103. 

First.  Let  the  pipe  be  open  to  the  atmosphere  at  B. 
Then 


h  —  head  to  overcome  resistance  to  entrance 


/  7' 

at  A  (=  n 
\        * 


Ci>^  \ 
=  m  —  J 

-f-  head  to  overcome  frictional  resistance  U=  -~  --  -) 

>       d   igl 


NOZZLES. 


'75 


+  head  corresponding  to  the  velocity  v  in  the  pipe  and 

(v*1  \ 
=  — 


FIG.  103. 
Hence  the  height  to  which  the  water  is  capable  of  rising 


at  B 


or,  again,  is 


=  ~~  =  h [n4-  m  +  ~-V 

- 


- 

Second.   Let  a  nozzle  be  fitted  on  the  pipe  at  B. 

Let  F  be  the  velocity  with  which  the  water  leaves  the 
nozzle. 

Let  D  be  the  diameter  of  the  nozzle-outlet. 

This  diameter  is  very  small  as  compared  with  the  diameter 
d  of  the  pipe.  But 


and  therefore 


_ 

v    '• —  • V. 

4  4     ' 


so  that  V  is  very  large  as  compared  with  v. 


1  76  NOZZLES 

Also, 
li  =  head  to  overcome  the  resistance  to  entrance  at  A 

+  head  to  overcome  the  resistance  due  to  bends,  etc. 

-f-  head  to  overcome  the  frictional  resistance  in  pipe 

-\-  head  to   overcome  the   frictional   resistance   in   nozzle 

i     '  v'\ 

\  —  m  - 
V  2f) 

-(-  head  corresponding  to  the  velocity  V  with  which  the 

'•      V2\ 
water  leaves  the  nozzle    I  —  .  —  j 


and  the  height  to  which  the  water  is  now  capable  of  rising  at 


h 
~&~ 

^~d* 

Let  —  ,  =  hn ,  be  the  pressure-head  at  the  entrance  to  the 
lozzle.      Then  the  effective  head  at  the  same  point 


7,2  J72 

=  h    A  --  =  (i  +m')—. 
r 


Hence 


It  will  be  observed  that  the  delivery  from  the  nozzle  is  less 
than  that  from  the  pipe  before  the  nozzle  was  attached,  but 
that  the  velocity-head  at  the  nozzle-outlet  is  enormously 
increased.  The  actual  height  to  which  the  water  rises  on 
leaving  a  nozzle  is  less  than  the  calculated  height,  owing  to 


NOZZLES. 


177 


air-resistance  and  to  the  impact  of  particles  of  water  as  they 
fall  back. 

The  force  required  to  hold  the  nozzle  is  evidently 


g  g    4 

If  the  water  flowing  through  a  pipe,  or  hose,  of  length  /  ft.  , 
with  a  velocity  of  v  ft.  per  second,  is  quickly  and  uniformly 
shut  off  by  a  stop-valve  in  t  sec.,  the  pressure  in  the  pipe  near 

wlv 
the  valve  is  increased  by  an  amount  -  Ibs.  per  square  foot. 

Of  two  forms  of  nozzle  in  general  use,  the  one  (Fig.  105) 
is  a  surface  of  revolution  with  a  section  which  gradually 
diminishes  to  the  outlet,  while  the  other  (Fig.  104)  is  a  frustum 


FIG.  104.  FIG.  105. 

of  a  cone,  having  a  diaphragm  with  a  small  circular  orifice  at 
the  outlet.  Denoting  the  former  by  A  and  the  latter  by  Br 
the  following  table  gives  the  results  of  Ellis's  experiments: 


Height  of  jet  from 
i-inch  Nozzle. 

Height  of  jet  from 
ij-inch  Nozzle. 

Height  of  jet  from 
ii-inch  Nozzle. 

Pressure  in  Ibs. 

Head  in 

per  sq.  in. 

feet. 

A 

B 

A 

B 

A 

B 

IO 

23 

22 

22 

22 

22 

23 

22 

20 

46 

43 

42 

43 

43 

43 

43 

30 

69 

62 

61 

63 

62 

63 

63 

40 

92 

79 

78 

81 

79 

82 

80 

50 

H5 

94 

92 

97 

94 

99 

95 

60 

138 

108 

104 

112 

1  08 

H5 

no 

70 

161 

121 

H5 

125 

121 

129 

123 

8c 

184 

131 

124 

J37 

131 

142 

135 

90 

207       1       140 

132 

148 

141 

154 

146 

100 

230 

148 

136 

157 

149 

164 

155 

The  coefficients  of  discharge  for  smooth  cone  nozzles  are, 
very  approximately,    .983   for  a  }-in.,    .982   for  a  ^-in.,  - 
for  a  i -in.,  .976  for  a  ij-in. ,  and  .971  for  a  ij-in.  nozzle. 


TABLE   OF  FRICTION AL   LOSSES  IN  HOSE. 
Freeman  proposed  the   ij-in.   nozzle  shown  by  Fig.    106 


FIG.  106. 

as  a  standard  with  a   coefficient    of  discharge  =  .977.      The 
coefficient  of  discharge  for  a  square  ring  nozzle  is  about  .74. 

FREEMAN'S      TABLE     SHOWING     COMPARATIVE     FRICTIONAL 
LOSS    IN    VARIOUS    KINDS    OF    HOSE. 

The  comparison  is  made  on  the  basis  of  a  flow  of  240  gals,  per  min., 
which  is  about  the  quantity  discharged  by  a  ii-in.  nozzle  under  a  pressure 
of  40  Ibs.  per  square  inch  at  base  of  play-pipe. 


r; 

U 

. 

^3 

3 

2  u 

£ 

bo 
c 

B 
rt 

J! 

e/3  U) 

S  = 

.52^ 

§1 

1> 

5 

*o£ 

0-«  §    . 

y"°   "  0 

•Sii« 

fc 

Cu 

.—  r 

qj    flj 

^  C-i    o 

3 

w 

c 

Character  of  Hose. 

O 

CJ 

CQ 

C     • 

ig| 

i'!« 

hij 

p 

0 

GK 

C  ^  3 

^  v  a 

O     U     V.     y 

OJ  T3   -J 

*-  u  5 

Q     g 

• 

I 

|° 

^t£ 

l^gl 

a||£ 

•sii 

^« 

c 

0*2"^ 

D    ^   ^* 

^  ^  *S  s> 

o)  0-~ 

C  u 

.2 

>  u 

OJ  3*0 

5  C  Q- 

UQ<5  5c 

S^Q 

S  a. 

Q 

* 

o'" 

c2 

,3 

s 

i\"  solid  rubber  hose,  extra  heavy,  smooth 

• 

and  free  from  ridges 

2  .  52" 

2  6c" 

/ 

IO.O 

i 

T1    ^ 

nfi 

25"  solid  rubber  hose  lighter  than  preceding 
and  not  so  carefully  made 

2.  53 

2.60 

+22 

aj"  woven  cotton  hose,  rubber-lined,  regular 

5 

' 

14  .0 

14  •  5^ 

heavy  fire-department  hose  . 

2     57 

15  .  o 

—   6 

Mr 
•  J 

rfi   m 

v\"  woven  cotton  hose,  rubber-lined,  lighter 

' 

*  •  47 

4 

' 

than  preceding,  but   of    about    the    same 

smoothness  of  interior  

2.47 

2  49 

5 

T4-5 

—    2 

14.2 

I5.8l 

•2\"  knit  cotton  hose,  rubber-lined.  A  medium- 

weight  hose  

2.50 

2.68 

3* 

JI-3 

+42 

16.0 

13.6"; 

az"  knit  cotton  hose,  rubber-lined.     Interior 

' 

medium  smooth 

2  .  5O 

2.50 

41 

16.8 

o 

16.8 

^O 

y\"  knit  cotton  hose,  rubber-lined.   A  regular 

4a 

' 

fire-department  hose.   .        .       .    . 
2j"  knit  cotton  hose,   rubber-lined.     Inside 

2-51 

2.60 

if 

13-9 

4-22 

17.0 

14.50 

rather  rough 

2.62 

_ 

•4-27 

18.3 

14.    28 

aj"  knit  cotton  hose,  rubber-lined.      About 

3 

*4  *4 

1    •*/ 

same  as  preceding,  but  a  little  heavier  .... 

2.51 

2.69 

27 

13-5 

-1-44 

TQ-4 

13  55 

2}"  leather  hose  

2.80 

2l 

12  .  2 

+76 

21     C 

aj"  woven  cotton,   rubber-lined,   mil!    hose. 

' 

Medium  thin  rubber  lining.  .  .    
2J"  unlined  linen  hose        

2    48 

2.50 

2-53 
2.60 

2^ 

24.1 
27.2 

4-  6 

+22 

25-5 

33-2 

15-  3' 
I4-5° 

2"  woven  cotton,  rubber-lined  hose  

2    07 

2.  12 

4* 

33  2 

-56 

14  6 

21.  8l 

si"  linen  hose  with  2"  couplings  

I-95 

2.30 

49  5 

-34 

32-7 

18.53 

WA  TER-MO  TOR.  1  7  9 

Third.  If  an  engine,  working  against  a  pressure  of  pc  Ibs. 
-per  square  foot,  pumps  Q  cu.  ft.  of  water  per  second  through 
a  nozzle  at  the  end  of  a  hose  /  ft.  in  length,  then 

Qpc 

the  pumping  H.P.  of  the  engine  =  -  . 

The  total  head  at  the  engine  end  of  the  hose  =  the  head 
corresponding  to  the  pressure  /  in  the  hose  +  the  head  required 
to  produce  the  velocity  of  flow  v 


W         2g- 

and  this  head  is  expended  in  overcoming  the  frictional  resist- 
ance of  the  hose  (all  other  resistances  are  disregarded)  and  in 
producing  the  velocity  of  flow  V  at  the  outlet.  Hence 

A=/  .  *=4/?«i  ,  F! 

W          W~^~  2g  d    2g  ~*~   2g  ' 

and  therefore 

/    _  '  4fl  v*        F2        v^ 

W~'     d     2g~\~  2g  2g' 


rtd*  TrD2 

since  Q  = v  =  -  —  F. 

4  4 

The  pumping  H.P. 

SwQ3    i  i 


550^ 


17.  Motor  Driven  by  Water  from  a  Pipe. — Let  the  nozzle 
in  the  preceding  article  be  replaced  by  a  cylinder  having  its 
piston  driven  by  the  water  from  the  pipe. 

Let  u  —  the  velocity  of  the  piston  per  second. 


i8o  EXAMPLE. 

Let/w  =  unit  pressure  at  the  end  of  the  pipe,  i.e.,  in  the 
cylinder. 

Let  dm  =  diameter  of  cylinder. 
Then 

Id  \2 
velocity  of  flow  in  pipe  =  f—j)  u. 

Hence 

i, f   fft\  ^-'    I   fft  \  i  •*  ** 

ftr      I  7      I  7    "  '          7~  > 

\  (I  /    2, g  Ci      \  d  I    2,g          "W 

other  losses  of  head  being  disregarded. 

Ex.  A  3i-in.  clean  pipe,  525  ft.  long,  leads  from  a  reservoir  with  a 
water  surface  300  ft.  above  datum  to  a  point  A,  187^  ft.  above  datum. 
Find  (a)  the  height  to  which  the  water  is  capable  of  rising  at  A  (i)  if 
the  pipe  is  open  to  the  atmosphere ;  (2)  if  it  terminates  in  a  i-in.  nozzle. 
What  (b)  force  is  required  to  hold  the  nozzle  ?  If  the  pipe  is  used  to 
supply  pressure  to  a  water-engine  with  a  28-in.  cylinder,  determine  (c) 
the  maximum  power  which  can  be  developed  and  the  corresponding 
velocity  of  flow  in  the  pipe.  In  the  latter  case,  what  (d)  is  the  total 
pressure  on  the  piston  ?  Take  into  account  the  resistance  at  the  pipe 
entrance  and  assume/  =  .005. 

Let  v  and  V  be  velocities  of  flow  in  pipe  and  from  nozzle,  respect- 
ively. 

(a)  i.  300  —  187^  =  1 12£  =  total  effective  head 

_    4  x  .005  x   5: 


3i 

12 

and  —  =  3  ft.  =  height  to  which  water  can  rise. 

d 

2-v  =  ~:  and 

II2.  =  £!  +  FV'  \V          4  x  .005  x  525\  _  V  2985 
11  **  —  -I  T -rriTr]  I  •>  T 


Therefore 

—  =  ii2£  x  — —  =  90.49  ft.  =  height  to  which  water  can  rise. 


SIPHONS. 


181 


Force  =  momentum 

62^    22     I/_LV/^a_  I25    ii      i       .,     22$    2401 

~32~'T   ~4V~i)          "64 '14   144'       '  ~ '  2985          Xt       s* 

Let  p  be  the  pressure  in  pounds  per  sq.  ft.  at  A.    Then 


or 


Hence 


theH.P.  =  Z*  = 


/  3_^2_2i/3_iv 
v     3r/7  4  Vi2y 


4 


55°          550 

=  {IP  —    7~1, 

3072^         64/' 

which  is  a  max.  when  3  —  ~-  =  o,      or     v  =  8  ft.   per  sec.,    and  the 

max.  H.P.   =  4^!  =  4.557. 

Also    p  =  62^ .  75  =  4687^  Ibs.  per  sq.  ft.,  and  total  pres.  on  piston 

=  46874  x  —.—  .[—]  —  ioTlk  tons. 

7     4     \i2j  2000 

18.  Siphons. — A  siphon  is  a  bent  tube,  A  BCD,  Fig.  107, 
and     is    often     employed    to 
convey  water  from  one  reser- 
voir   to    another    at   a    lower 
level. 

Let  h^ ,   h2 ,    respectively, 
be    the    differences     of    level 
between  the  top  of  the  siphon 
and  the  entrance  A  and  outlet 
D  to  the  siphon.      Then,   so 
long  as  the  height  h^  does  not 
exceed     the    head    of    water    „______ 

{=  32.8  ft.)  which  measures  FlG-  ID7- 

the  atmospheric  pressure,  the  water  will  flow  along  the  tube  in 

the   direction   of  the   arrow,   with  a  velocity  v  given  by  the 

equation 


182 


INSERTED  SIPHONS. 


I  being  the  length  of  the  tube  A  BCD,  and  all  resistances, 
except  that  due  to  frictional  resistance,  being  disregarded. 

If  //j  >  32.8  ft.,  each  of  the  branches  AB  and  DC  becomes 
a  water-barometer,  and  the  siphon  will  no  longer  work. 

Even  when  the  siphon  does  work,  an  arrangement  must  be 
made  for  withdrawing  the  air  which  will  always  collect  at  the 
upper  part  of  the  siphon. 

19.  Inverted  Siphons. — The  existence  of  a  cutting  or  a 
valley  sometimes  renders  it  necessary  to  convey  the  water  from 
a  course  AB  to  a.  course  DE  by  means  of  an  inverted  siphon 
BCD  of  length  /. 

Let  u  be  the  velocity  of  flow  in  AB,  and  /^  the  height  of  B 
above  a  datum  line. 

Let  v  be  the  velocity  of  flow  in  the  siphon,  and  7z2  the 
height  of  D  above  datum. 


FIG.  108. 


Then 


hv  —  h2  =  loss  of  head  at  B 

+  frictional  loss  of  head  in  siphon 
-f-  loss  of  head  at  D 

—  ^!  \^IL—  \  — 

~  2g    '       d    2g  ~"~  2g 

=  —j ,  approximately, 

assuming  the  entrance  and  outlet  to  the  siphon  formed  in  such 

iP  v* 

a  manner  as  to  considerably  reduce  the  losses  —  and  — ,  and 


AIR  IN  A  PIPE.  183 

to  allow  of  these  losses  being  disregarded  without  practical 
error.  Find,  by  chaining  along  the  ground,  the  length  of  the 
siphon  from  B  up  to  a  point  F  not  far  from  D.  Call  this 
length  /j ,  and  let  //3  be  the  height  above  datum  of  F,  obtained 
with  a  level.  Generally  speaking,  DF  is  nearly  always  of 
uniform  slope.  Call  the  slope  a.  Then, 


DF  = 


But 


—        cosec  a. 


=  hl  —  h^  —  DF .  sin  a, 

an  equation  from  which  DF  can  be  found,  as  h^  —  /i3  can  be 
determined  by  means  of  a  level. 

20.  Air  in  a  Pipe. — The  effect  of  an  air-bubble  in  a  pipe 
ABCD  may  be  discussed  as  follows: 

Let  the  air  occupy  the  portion  BC  of  a  pipe. 

Let  the  surface  of  the  water  in  the  reservoir  supplying  the 
pipe  be  h^  ft.  vertically  above  £,  and  /z2  ft.  above  D. 


FIG.  109. 

Also,  let  h^  be  the  difference  of  level  between  C  and  D,  h± 
the  difference  of  level  between  B  and  C,  and  /  the  thickness  of 
the  water-layer  EF. 

Let  //"designate  the  head  equivalent  to  the  elastic  resistance 
of  the  air  in  BC.  Then,  approximately, 

P  4./^i  V* 

1  +  w  '  ~d    ~2g 


i84 
and 


FLOW  IN  PIPE  OF  VARYING  DIAMETER. 


w 


-=  -- 
d    2g 


/!  being  the  length  of  the  portion  of  pipe  from  A  to  E,  and 
/2  the  length  from  E  to  D. 
Adding  the  two  equations, 


/  being  total  length  of  pipe. 
But  h^  —  t  + 


^  ,  very  nearly.      Hence 


an  equation  showing  the  variation  of  v  with  a  variation  in  the 
height  h±  of  the  space  occupied  by  the  air. 

NOTE.  —  H  of  course  varies  with  the  temperature. 
21.  Flow  of  Water  in  a  Pipe  of  Varying  Diameter.  —  The 
variation  in  the  diameter  is  supposed  to  be  so  gradual  that  the 

fluid  filaments  may  still  be  assumed 
to  flow  in  sensible  parallel  lines. 

Consider    a    thin    slice   of    the 
moving  fluid,  bounded  by  the  trans- 
verse sections  AB,  CD,   distant  s 
and  s  +  dsj  respectively,  from  an 
~  origin  on  the  axis  of  the  pipe. 

Let  /  be  the  mean  intensity  of 
pressure,  A   the  water  area,  P  the 
wetted  perimeter  for  the  section  AB. 
Let  these   symbols   become  /  +  dp,    A  +  dA,    P  +  dP, 
respectively,  for  the  section  CD. 

Let  z  be  the  height  of  the  C.  of  G.  of  the  section  AB  above 
datum. 

Let  z  -f-  dz  be  the  height  of  the  C.  of  G.  of  the  section  CD 
above  datum. 


FIG.  no. 


FLOW  IN  PIPE   OF  VARYING  DIAMETER.  185 

Let  uy  u  +  du  be  the  velocities  of  flow  across  the  sections 

,  CD,  respectively. 

Then 


The  rate  of  increase  of 
momentum  of  the  slice 
ABCD  in  the  direction 
of  the  axis 


momentum  generated  by 
the  effective  forces  acting 
upon  the  slice  in  the  same 
direction. 


iv  du       w 

The  acceleration  in  time  dt  =  —  Au  .  dt~  =  —Au  .  du. 

g  dt         g 

The  total  pressure  on  AB  =  /  .  A,  and  acts  along  the  axis. 
The    total   pressure   on    CD  —  (p  +  dp)(A  +  dA),    and   acts 

along  the  axis. 
The  total  normal  pressure  on  the  surface  A  CBD  of  the  pipe 

(dr\  I          dp  \ 
r  -\  --  )!/+  —  \AC  —  2nrp  .  AC,  very  nearly. 

The  component  of  this  pressure  along  the  axis 

=  2  nrpA  C  .  sin  6 
=  2  npr  .  dr,  nearly, 

6  being  the  angle  between  AC  and  the  axis. 

Thus  the  total  resultant  pressure  along  the  axis 
—  pA  —  (p  -f.  dp)(A  -f  dA)  +  2npr  .  dr 
=  —  p  .  dA  —  A  .  dp  -j-  2  npr  .  dr 
=  -A.dp, 
since  A  =  nr2,  and  therefore  dA  —  2nr  .  dr. 

The  component  of  the  weight  ^/"the  slice  along  the  axis 


=  f  A  H  --  \ds  .wsmi—  --  (  A  -\  --  \w  .  dz  =  — 


wA  .  dz. 


The  frictional  resistance  =  P  .  AC  .  F(u)  =  P  .  ds  .  F(it), 
very  nearly.      Hence 

-  =  —A  .  dp  -  wA  .  dz  —  P  .  ds  .  F(u\ 


1 86  EQUIVALENT  UNIFORM  MAIN. 

and  therefore 


,     .dp     ,    u.  du        P  F(u\ 

dz-\-~-  A \-  —  — "-tds  —  o. 

w  g        '    A     w 


Integrating, 


.      /         ««          C  P  F(u), 

'  H H +   /   ^ »»  =  a  constant. 

1   w        2g   '  J  A     w 


Talr<=»  f 

±  clKC  / 

W  2g  2g 

Then 


W          2g         /    gn< 


=  a  C0nstant 


The  integration  can  be  effected  as  soon  as  the  relation 
between  r  and  s  is  fixed. 

Example. — Take  r  =  a  +  bsy  and  assume  f  and  Q  to  be 
constant.  Then 


/    ,    u*     ,    i/(?     Cdr 

— \-j ^o    I  -r  = 

W      ig       b  gti  J    r5 


^  +  -.  +  T^  +  r  T±2    /  7T   :  =  a  constant, 

and  therefore 

p        u*         i    /£>2    i 
^  +  ~-  H h  -T  -^  -i  =  a  constant. 

22.  Equivalent  Uniform  Main. — A  water-main  usually 
consists  of  a  series  of  lengths  of  different  diameters. 

As  a  first  approximation  the  smaller  losses  of  head  due  to 
changes  of  section,  etc.,  may  be  disregarded,  and  the  calcula- 
tions may  be  further  simplified  by  substituting  for  the  several 
lengths  a  single  pipe  of  uniform  diameter  giving  the  same  fric- 
tional  loss  of  head.  Such  a  pipe  is  called  an  equivalent  main. 


EQUIVALENT  UNIFORM  MAIN.  187 

Let  /! ,  /2 ,  /3  be  the  successive  lengths  of  the  main. 


FIG.  in. 

Let  dlt  dZJ  d^  be  the  diameters  of  these  lengths. 

Let  vl,  v2,  v^  be  the  velocities  of  flow  in  these  lengths. 

Let  ^,   ^2,   h^  be    the   frictional   losses   of  head  in  these 

lengths. 
Let   L,    d,   v,  h  be   the   corresponding   quantities   for   the 

equivalent  uniform  main. 

Then 


and  therefore 


Hence 


_ 
rf, 


TV     ^3   ?      i 
^7"  ^TI.  3  ~1 


where  it  is  assumed  that  /is  the  same  for  the  several  lengths 
of  the  main  and  also  for  the  equivalent  pipe. 
But 


nd* 
— 


nd 


Hence 


an  equation  giving  the  diameter  d  of  an  equivalent  pipe  having 
the  same  total  frictional  loss  of  head. 


1  88  BRANCH  MAIN  OF  UNIFORM  DIAMETER. 

Ex.  What  must  be  the  diameter  of  a  uniform  pipe  which  may  be 
substituted  for  a  line  of  piping  consisting  of  an  8oo-ft.  length  of  12-in. 
pipe  and  a  2Oo-ft.  length  of  6-in.  pipe? 

800  +  200        800      200 

-y-  =  -F-  +  a/  =  720Dl 

<*'  =  3i 

and  therefore  d  =  .6738  ft.,  or  about  8  ins. 

23.  Branch  Main  of  Uniform  Diameter.  —  In  a  branch 
main  AB  of  length  L  and  diameter  d,  receiving  its  supply 
at  A,  — 

Let  Qw  be  the  way-service,  i.e.,  the  amount  of  water  given 

up  to  the  service-pipes  on  each  side. 
Let   Qe  be  the  end-service,  i.e.,  the  amount  of  water  dis- 

charged at  the  end  B. 
Then  it  may  be  assumed,  and  it  is  approximately  true,  that 

the  way-service  per  lineal  foot,  viz.  ,  =~  ,  is  constant. 

Thus  the  amount  of  water  consumed  in  way-service  in  a 
length  AC  of  the  main,  where  BC  =  sy  is 


while  the  total  amount  of  water  flowing  across  the  section  of 
the  pipe  at  C 


-L  4 

v  being  the  velocity  of  flow  at  C. 

Now  dh,  the  frictional  loss  of  head  at  C  for  an  elementary 
length  ds  of  the  pipe,  is  given  by  the  equation 


d  2g 


SPECIAL   CASES  OF  PIPE-FLOW.  189. 

Integrating,  the  total  loss  of  head  is 


SPECIAL    CASES. 

CASE  I.  Let  Qe'  be  the  total  discharge  for  the  same  fric- 
tional  loss  of  head,  h,  when  the  whole  of  the  way-service  is 
stopped.  Then 

^O'*=h  = 


and  therefore 


Hence 

Q.' 2  >  (O.  +  % )'     and       <  (fi,  +  ^, 

and   QJ  lies  between  Qe  -j-  ~^  and  Qe  -| — JL ,  its  mean  value 

being  Qe+.S$Qw. 

CASE  II.      If  there  is  no  end-service,  all  the  water  having 

been  absorbed  in  way-service,  Qe  —  o,  and  therefore  Qe'  —  — = 

and 

T  fT  O  2 
h  = 


CASE  III.     If  Qt  =  o, 

fO  2 
dk  =    9  -,^T  9s*ds  =  elementary  frictional  loss  of  head. 

52  * 


SPECIAL   CASES   OF  PIPE-FLOW. 
Integrating  between  o  and  j, 


and   the  vertical   slope,    or   line  of  free   pressure,   becomes    a 
cubical  parabola. 

CASE  IV.  Let  the  main  receive  its  supply  at  A  from  a 
reservoir  X  in  which  the  surface  of  the  water  is  hl  above 
datum,  and  let  it  discharge  at  the  end  B  into  a  reservoir  Y  with 
its  surface  //.,  above  datum,  Fig.  114. 

Since          (0/)a  =  a2  +  QeQw  +  ^,  therefore 


<2,  =  o;  and  if  Gw>  ^3<2/«  ^en  the 
reservoir  K  will  furnish  a  portion  of  the  way-service. 

Suppose  that  X  gives  the  supply  for  the  distance  AO  (=  /L) 
and  that  Y  supplies  BO  (=  /2). 

Let  z  be  the  height  above  datum  of  the  surface  in  a  pressure 
column  inserted  at  0. 

Then,  neglecting  the  loss  of  head  at  entrance, 


2/3 
=  loss  of  head  between  A  and  <9  =  - 


and 


i  fO  2ts 

=  loss  of  head  between  .#  and  6>  =  -  J  ^  *. 

3  7f*d*L* 


Also,  /i  +  /,  =  /:. 


PROBLEM  OF   THREE  RESERVOIRS. 


191 


24.  Three  Reservoirs  at  Different  Levels  connected  by  a 
Branched  Pipe. — Let  a  pipe  DO  of  length  !{  ft.  and  radius 
rl  ft.,  leading  from  a  reservoir  A  in  which  the  water  stands 
//t  ft.  above  datum,  divide  at  O  into  two  branches,  the  one, 
OE,  of  length  /.,  ft.  and  radius  r.,  ft. ,  leading  to  a  reservoir  B 
in  which  the  water  stands  //2  ft.  above  datum,  the  other,  OF, 
of  length  /3  ft.  and  radius  rs  ft. ,  leading  to  a  reservoir  C  in 
which  the  water  stands  h.,  ft.  above  datum. 


FIG.  ii2. 

Let  i\ ,   v2 ,   ^3  be  the  velocities  of  flow  in  DO,  OE,  OF, 

respectively. 
Let   Q^  Q2,  Q3  be  the  quantities  of  flow  in  DO,  OE,  OF, 

respectively. 

Let  z  be  the  height  above  datum  to  which  the  water  will 
rise  in  a  tube  inserted  at  the  junction. 

Two  problems  will  be  considered,  and  all  losses  of  head 
excepting  those  due  to  frictional  resistance  will  be  disregarded. 

PROBLEM  I.      Given  //,,  //2,  h^  r^,  r2,  r3;  to  find  Qlt  Q2, 
£>3;  vlt  v2,  vs,  and  s.      Taking  -  =  a, 

o  * 


192  PROBLEM  OF  THREE  RESERVOIRS. 

For  the  pipe  OE,^f^=c&  .  .  (3)    "     Q1  =  xr2\.  .  .  (4) 

12  r2 

"      "     "      OF,    £=*•=*£..  (S)     ..    'Q^Krfr.  .  .  (6) 

*3  *3 

Also,  0,=  ±<22+<23  .......     (7) 

From  these  seven  equations  the  seven  required  quantities 
can  be  found. 

In  equations  (3)  and  (7)  the  upper  or  lower  signs  are  to 
be  taken  according  as  the  flow  is  from  O  towards  E  or  from  E 
towards  0. 

This  may  be  easily  determined  as  follows  : 

Assume  z  =  h2  ,  and  then  find  vl  and  vz  by  means  of  equa- 
tions (i)  and  (5),  and  hence  Ql  and  Q3  by  means  of  equations 
(2)  and  (6).  If  it  is  found  that  Ql  >  gs>  then  the  flow  is  from 
0  to  E,  and  equations  (3)  and  (7)  become 

'-="-  and    ft  =  ft'+^; 


while  if  (2i  <  GS  >  the  fl°w  is  from  £  to   6>,  and  the  equations 
are 

7  2  * 

-L-^lA     and  =       . 


.  —  It  is  assumed  that  «f=  —  J  is  the  same  for  each  pipe. 

<*> 

SPECIAL  CASE.    (Fig.  113.)  —  Suppose  the  pipe  OE  closed 

Bt£. 

Also,  let  rl  =  r2  =.  r3  —  r,  and  let  V  be  the  velocity  of  flow 
from  A  to  (7. 

The  *  l  plane  of  charge  '  '  for  the  reservoir  A  is  a  horizontal 

plane   MQ  distant  ~   from    the    water-surface,   /0    being  the 
atmospheric  pressure. 


V 


PROBLEM   OF   THREE  RESERVOIRS.  193 

The  "  plane  of  charge  "  for  the  reservoir  C  is  a  horizontal 

plan-e  TS  distant  —  from  the  water- surface. 
w 

F2 
In  the  vertical  line    VTQ,   take    TN  =  —  and  join  MN. 

o 

Then,  neglecting  the  loss  of  head  at  entrance,  MN  is  the 
4 *  line  of  charge,"  or  hydraulic  gradient,  for  the  pipe  DF,  and 
is  approximately  a  straight  line. 

Let  the  ' '  plane  of  charge  ' '  KK  for  the  reservoir  B,  distant 

—  from  the  water-surface,  meet  MN  in  G. 

w 

If  the  junction   0  is  vertically  below  6",  there  is  no  head" 


FIG.  113. 

available  for  producing  flow  either  from  E  towards  O  or  from 
O  towards  E,  and  hydrostatic  equilibrium  is  established. 

If  the  junction  O  is  on  the  left  of  G,  and  a  vertical  line 
OKHL  is  drawn  intersecting  KK,  MN,  and  MQ  in  the  points 
K,  //,  and  Z,  there  is  the  head  HK  available  for  producing 
flow  from  O  towards  E. 


194  PROBLEM  OF   THREE  RESERVOIRS. 

If  the  junction  O  is  on  the  right  of  G,  and  the  vertical  line 
OHKL  is  drawn,  the  head  HK  is  now  available  for  producing; 
flow  from  E  towards  O. 

Let  the  vertical  through  G  meet  MQ  in  P,  and  take 
PG  =  Y.  Then,  approximately, 

/  MG       PG  F 


QN 
and  therefore 


If  HL  <  F,  the  flow  is  from  O  towards  E. 
If  HL  >  F,  "  "  "  "  £  "  O. 
Again, 


and  therefore,  approximately, 


(O 


Next  assume  the  junction  O  to  be  on  the  left  of  G,  and 
open  the  valve  at  E.      Then 


(2> 


^  =  4; (3> 


(4) 


ORIFICE  FED  BY   TWO  RESERVOIRS.  19$ 

and  G1=(22+G8, 

or  *  vl  =  7'2 '+  7/s. 

Thus 

«7<4+4)  -  *!-*•=  "(A",2  +  W)  =  7  j  4("2  +  "JM- W }  ; 

and  therefore 

»,V,  +  4)  +  2'w,  +  to'  -  d  +  4)  ^2  =  o. 

Hence,  assuming  z-2  to  be  very  small  as  compared  with  F, 


or 


where  Q  =  rrr2  V, 

Thus  it  appears^that  if  a  quantity  Q2  of  water  is  drawn  off" 
by  means  of  a   branch  from  a  main  capable  of  giving  a  total 
end-service  Q,  this  end-service  will  be  diminished  by  j-(22,  i<2., , 
J<22,  etc.,  according  as  the  junction  O  divides  the  pipe  DF  into 
two  portions  in  the  ratio  of  I  to  I,   I  to  2,   I  to  3,  etc. 

NOTE. — The  more  correct  value  of  i>   is 

o 


and  the  maximum  value  of  71 — I-—-.-,  does  not  exceed  -. 

(A  -t-  4)  4 

Orifice  Fed  by  Two  Reservoirs. — Neglect  all  losses  of  head 
except  the  losses  due  to  frictional  resistance. 

When  the  valve  at  O  is  closed  the  flow  is  wholly  from  A 
to  Cy  and  the  delivery  is 


= 


ORIFICE  FED  BY   TWO  RESERVOIRS. 
The  line  of  charge  (hydraulic  gradient)  is  MN,  where 


=-^  =  NV. 

w 


FIG.  114. 

Open  the  valve  a  little:  a  volume  Q.z  will  now  flow  through 
,  and  a  volume  <23  into  C,  where 


a  =  Q  - 


.  + 


The  "  line  of  charge  "  becomes  the  broken  line  M\N. 

As  the   opening  of  the  valve  continues,  the  pressure-head 

at   O  diminishes,  and  when   it  is  equal  to  //3-[-^  -  the  line  of 

charge  [$M2N\  2N  being  horizontal.  Hydrostatic  equilibrium 
Is  now  established  between  O  and  Ct  and  the  whole  of  the 
water  from  A  passes  through  O,  the  delivery  being  given  by 


^Opening    6^    still    further,    both    reservoirs   will    serve   the 
corifice,  and  the  line  of  charge  will  continue  to  fall. 


EXAMPLE. 

When  the  valve  is  full  open  the  '  '  line  of  charge  '  '  is 
\vhere     6*  = 


197 


,  and  the  discharge  is 


The-  supply  from  A  is  equal  to  that  from  C  when  ~  —  -~. 

A        ^3 

The  above  investigation  shows  the  advantage  of  a  second 
reservoir  in  emergent  cases  when  an  excessive  supply  is 
suddenly  demanded,  as,  e.g.,  on  the  occasion  of  a  fire. 

Ex.  A  <24-in.  pipe  AB,  6000  ft.  long,  connects  two  reservoirs,  the  dif- 
ference of  level  between  the  water-surfaces  being  250  ft.  From  a  junc- 
tion O  between  A  and  B  a  12-in  pipe,  C,  3000  ft.  long,  connects  with 
an  intermediate  reservoir  having  its  water-surface  150  ft.  above  that  of 
the  lowest  reservoir.  Discuss  the  distribution  (a)  when  AO  =  2000  ft. ; 
(£)  when  AO  =  4000  ft.;  and  find  (c)  the  position  of  O  so  that  there  ma- 
be  no  flow  in  OC, 


FIG.  115.  FIG.  116. 

Take  the  lowest  water-surface  as  the  datum  plane.    Also  assume  that 
a  —  J-  —  .0002.  r 


19$  EXAMPLE. 

If  a  piezometer  is  inserted  at  O,  the  water  will  rise  in  it  to  a  height 
2  above  datum. .  Then 
(a)  Fig.  115: 

Between  A  and  O 

250  —  z 

2000 
Between  O  and  B 

±    I  50  T  Z  __         7',5  _  a 

3000  ^    ~ 

Between  O  and  C 

Z  "3* 

=  a  — =  acvi. 


(I) 


4000  I 

To  find  the  direction  of  the  flow  in  OC,  let  z  =  150,  then  z/a  =  o, 
av*  —  ^,  ar/39  =  -£\,  and  therefore  7/1  >  7/3.  Thus  more  water  flows 
from  A  to  O  than  is  required  for  the  lowest  reservoir,  and  a  portion 
must  flow  to  the  intermediate  reservoir.  Hence  z  >  150  ft.,  and 


Therefore 


~  * 


20000:         4  '       6ooo«  40000: 


or  |/7i5oo  —  62  =  —  \/2z  —  300  +  Vy. 

By  trial  this  gives  z  =  161   ft.,  very  nearly,   and  then,   substituting 
in  eqs.  (I), 

7/1'  ==  222.5,    or    ^i  =  14.916  ft.  per  sec., 

W  =  9i,          or    v.,  =    3.027  " 

7/3a  =  201.25,  or    7'3  —  14.186  "        " 


Hence,  also, 


22       22 

,  =  —  .  —  x   14.916—  46.879  cu.  ft.  per  sec., 
7      4 

'  =      '     x   3'027=  2'378    " 


3  =        .      x  14.186  -44-584     " 
7      4 


and  £>„  +  gs  =  46.962  =  Qi,  very  nearly. 


EXAMPLE.  199 


Fig.  1  16: 
Between  A  and  0 

250  —3 

-^  -        = 
4000 

Between  O  and  B 


3000 

Between  (9  and  C 


(ID 


2OOO 

If  2  —  150,  7/2  =  o,  avi*  =  ¥V»  and  ttz/s2  =  A-  Tnus  ^  >  ^  and  there- 
fore Q3  >  Qi,  so  that  more  water  flows  to  the  lowest  reservoir  than  is 
•supplied  by  the  highest  reservoir.  Hence  the  balance  must  come  from 
the  intermediate  reservoir  and  z  <  150  ft. 

AISO,  0i    +   <2a   =   03  > 


7.'a 
+  --   =  7/s. 


Therefore 


40000:  4  0000  2000« 


or  -1/1500  —  62  +  —  1/300  —  22  =  }/6z. 

4 

By  trial  this   gives    z  —  96,  very   nearly,   and   then,  substituting  in 
<eqs.  (II), 

7/!2  =:  192.5,    or     2/1  =  13.874  ft.  per  sec., 
TV  =  45,         or     7/2  =    6.709  "         " 
7/39  =  240,       or     7/3  =  15.492  " 
Hence,  also, 

_  22    2_    x   ^g^     =  43.604  cu.  ft.  per.  sec., 
7  '4 

0,  =  —  .!  .  i2  x  6.709=    5.271       " 

0,  =  —  .  — .  2s  x  1 5.492  =  48.689      "  "  =  0i  +  0a  very  nearly. 

(c)  Let  AO  =  x.     Then,  since  7/2  =  0,  z  —  150  ft.,  and  therefore 
250  —  150  _    ^  2  _        2  _        150 

*  ~  6000^? 

Hence 

— r  =  —  =  — ,     and    ^r  =  2400  ft. 

100       2 


2QQ  PROBLEM  OF  THREE  RESERVOIRS. 

PROBLEM  II.      Given  /tlt   h2,   h^\    Q2,    Q3,   and  therefore 

Ql  (=  ±  G2  +  Gs);  to  find  ri>  r*>  ^3>  *'i»  Vj.'^s*  '*• 

As  before,  let  s  be  the  pressure-head  at  O.      Then 


—  2 


,  ,  , 

'-—  '=tf_L    .      .      .     (i)     and      0,=  *^,;        .     .      (2) 
/i  .  rj 


4  ^2 


•      •      •      (3)       "        <22  =  *rfvt;       .      .      (4) 


These  six  equations  contain  the  seven  required  quantities,, 
viz.,  fj,  r2,  fs,  z/j,  ^2,  ^3,  and  ^.  Thus  a  seventh  equation 
must  be  obtained  before  their  values  can  be  found.  This 
equation  is  given  by  the  condition  '  '  that  the  cost  of  the  piping- 
laid  in  place  should  be  a  minimum,"  it  being  assumed  that  the 
cost  of  a  pipe  laid  in  place  is  proportional  to  its  diameter. 

Hence 

llrl  -\-  /2r2  -|-  /3r3  =  a  minimum.     •  .      .      .      (7) 


From  equations  (i)  and  (2), 


"(6), 


Differentiating  these  three  equations, 
dz        laQ? 


2 


—        a      5> 


/f  "       "    *r*    ' 


MAINS  WITH  SEVERAL  BRANCHES.  2O£ 

But,  by  equation  (7), 


Hence 


which  is  the  seventh  equation  required. 

This  last  equation  may  be  written  in  the  forms 


and 

e._  ,0.4.  Q, 

V"  t*,'  +  ».r 

25.  Mains  with  any  Required  Number  of  Branches. 

Let  there  be  n  junctions  and  m  pipes. 

Let  ^,  Aj,  .  .  .  hm  be  the  m  pressure-heads  at  the  end  of 

each  successive  length  of  pipe. 
Let  zlt  z2,  .  .  ,  zn  be  the  n  pressure-heads  at  the  1st,  2d, 

3d,  .  .  .  nth  junctions. 

Let  /n  /2 ,.../»»  be  the  lengths  of  the  m  pipes. 
PROBLEM  I.     Given  klt  //2,  .  .  .  hm ,  rlt  r2,  .  .  .  rm\  to 
find  vlt  v2,  .  .  .  vm1  zl,  z2,  .  .  .  ZH. 

±  k  -^  z  V* 

There  are  m  equations  of  the  type  -. =  a— . 

Also,  the  quantity  flowing  through  the  first  portion  of  the 
main  is  equal  to  the  sum  of  the  quantities  flowing  through  all 
the  branches  at  the  first  junction,  and  an  analogous  equation 
will  hold  for  each  of  the  remaining  n  —  I  junctions.  Thus  n 
additional  equations  are  obtained. 

From  these  m  -J-  n  equations  7^  7>2 ,  .  .  .  vm ,  sl ,  z2, 
.  .  .  zn  may  be  found  analytically  or  by  the  method  of 
repeated  approximation. 


202       VARIATION  OF  VELOCITY  IN.  TRANSVERSE  SECTION. 

PROBLEM  II.     Given  /ilt  /i2,  .  .  .  //,„,  Ql ,  Q2,  .  .  .  Qm\  to 
find  rlt  r2,  .  .  .  rm,  ^ ,  z^  .  .  .  zn. 

There  are  now  only  m  equations  of  the  type 


±  h 


/ 


=  a— 
r 


involving  m  -(-  n  unknown  quantities,  and  the  problem  admits 
of  an  infinite  number  of  solutions. 

It  is  therefore  assumed  that  the  cost  of  the  piping  laid  in 
place  is  to  be  a  minimum.  Thus  n  new  equations  are  obtained, 
and  the  m  -\-  n  equations  may  be  solved  analytically  or  by 
repeated  trial. 

NOTE.  —  The  maximum  velocity  of  flow  in  town  mains  is 
from  2  to  7  ft.  per  second. 

26.  Variation   of  Velocity  in   a  Transverse  Section.— 
Assumption.  —  That  the  water  in  any  portion  of  a  pipe  is  made 
up  of  an  infinite  number  of  hollow  concentric 
cylinders   of  fluid,  each    moving  parallel  to 
the  axis  with  a  certain  definite  velocity. 

Let    u    be  the  velocity  of  one    of  these 
cylinders    of  radius    x   and     thickness    dx. 
Then   the  flow   across   a   transverse  section 
FIG.  117.          *~  is  given  by  the  equation 


dq  =  27tx  dx  .  uy 


and  the  total  flow 


(i) 


r  being  the  radius  of  the  pipe. 

If  vm  be  the  mean  velocity  for  the  whole  transverse  section 
of  the  pipe, 


Q 


(2) 


VARIATION  OF  VELOCITY  IN   TRANSVERSE  SECTION.      203 

Again,   assuming  with   Navier  that  the  surface  resistance 
Between  two  concentric  cylinders  is  of  the  nature  of  a  viscous 

resistance  and  may  be  represented  by  k —  per  unit  of  area  at 

the  radius  x,  k  being  a  coefficient  called  the  coefficient  of 
viscosity,  then  the  total  resistance  at  the  radius  x  for  a  length 
ds  of  the  cylinder 

.du  du 

=  —  2n x  .  ds  .  k-j-  =  —  2nk  .  ds  .  x-j-. 
dx  dx 

The  total  resistance  at  the  radius  x  +  dx 

r  du        d(  du\  ,-\ 
—  _j_  2nk  .  ds\  x  —  -j-  -j-\x-r  }dx    . 
L  dx*    dx\  dxl      J 

Hence  the  total  resultant  resistance  for  the  length  ds  of  the 
cylinder  under  consideration 


d  I  du\ 
=  2nkds-r\x-r\dx. 

dx\  dx}  . 


The  component  of  the  weight  of  the  slice  of  the  cylinder 
in  the  direction  of  the  axis 

=  w  .  2  nx  .  dx  .  ds  .  sin  0, 

6  being  the  inclination  of  the  axis  to  the  horizon. 

Let  —  dz  be  the  fall  of  level  in  the  distance  ds.      Then 

—  ds  =  ds  .  sin  B.    • 
Therefore  component  of  weight  in  direction  of  axis 

=  —  w  .  2  xx  dx  .  ds. 


204       VARIATION  OF  VELOCITY  IN   TRANSVERSE  SECTION. 

The    resultant    pressure    on    the   slice  in    the   direction   of 
motion 

=  \P  —  (f  +  dp)\2nx  .  dx  =  —  2nx  .  dx  .  dp. 
Then,  since  the  motion  is  uniform, 

w  .  2  nk  .  ds  .  ~T~(^~j-)dx  —  w  .2nx  .  dx  .  dz—2nx  .  dx  .  dp  —  O, 

and  therefore 

k  .  ds  d  I  du\  dp 

-j-  1  *  j   1  —  dz  --  -  =  o. 


x     dx\  dx)  w 

Integrating  only  for  the  cylinder  under  consideration, 

ks  d  I  du\        I        p\ 

~~j~\x~r  )  —  \s  -4-  —    =  a  constant. 
x  dx\  dxl        \     v  w) 

But  z  -f-  —  is  evidently  independent  of  x    and  is  a  linear 
function  of  s  (Art.   5,  Chap.  II).      Hence 

i    dL  du\ 

— T\XT~\  =  a  constant  =  A*  suppose. 

x  dx\  dxl 

Therefore 


Integrating, 


dx 


Assuming  that  the  central  fluid  filament  is  the  filament  of 
maximum  velocity,  then  when  x  =  o,  -j-  is  also  nil.     Therefore 

du       Ax* 

B  =  o,      and     x-j-  =  -  —  , 
dx         2 


VARIATION  OF  VELOCITY  IN   TRANSVERSE  SECTION.      205 

• 

and  therefore 

du  * 

Z=:A2 -..     (4) 

Integrating,  Eq.  4, 


C  being  a  constant  of  integration. 

Since  dp  is  the  difference  of  intensity  of  pressure  on  the 
ends  of  the  cylindrical  slice, 

du 
—  2nx  .  ds  .  k-r  =  nx*  .  dp  —  7tx*w  .  dh. 

Therefore 

du  ivx  dh  wxi 

dx  ~          2k  ds  2k1 

and,  by  equation  (4), 

wi 


Let  «max  be  the  velocity  of  the  central  filament,  i.e.,  the 
value  of  u  when  x  =  o.      Then 


and 

«.«.-*=    --x*  =  Dx\      .      .     .     .     (5) 


where  D  = • 

4 

Again,  by  equation  (i), 

.dx 


206       VARIATION  OF  VELOCITY  IN   TRANSVERSE  SECTION. 
and  by  equation  (2), 

Dr* 


(6) 


If  w,  =  velocity  at  pipe  wall,  then,  by  equation  (5), 

«*  =    ^inax.  -  /V  ......        (7) 

Hence,  by  equations  (6)  and  (7), 

U5  +  fcmax.  =    *VM.  .....         (8) 


If  u  =  o  when  x  —  ry  then  C  =  —  A— ,    and 

4 


—K- 

Therefore 


Anr*       winr* 


NOTE. — In  a  paper  by  Gardner  S.  Williams  and  others,  in  the  Pro- 
ceedings of  the  Am.  Soc.  of  C.  E.  for  May,  1901,  giving  the  results  of 
experiments  on  the  flow  of  water  in  pipes,  the  inferences  are  made  . 
that  at  ordinary  velocities  of  flow,  and  under  normal  conditions,  the 
ratio  of  the  mean  velocity  to  the  maximum  13.84;  that  in  a  straight 
pipe  there  will  be,  under  some  conditions,  a  difference  of  pressuie  at 
different  points  in  the  circumference  of  the  same  cross-section  ;  that 
the  normal  curve  of  velocities  is  an  ellipse;  that  the  effect  of  a  flow 
disturbance  is  felt  many  diameters  beyond  the  point  at  which  it  occurs ; 
that  for  a  maximum  flow  careful  alignment  is  as  necessary  as  a 
smooth  interior. 


WATER-METERS. 


207 


27.  Gauging  of  Pipe-flow. — A  variety  of  meters  have 
been  designed  to  register  the  quantity  of  water  delivered  by  a 
pipe.  The  principal  requisites  of  such  a  meter  are : 

1.  That  it  should  register  with  accuracy  the   quantity  of 
water  delivered  under  different  pressures. 

2.  That  it  should  not  appreciably  diminish   the    effective 
pressure  of  the  water. 

3.  That   it   should   be    compact   and    adaptable    to   every 
situation. 

4.  That  it  should  be  simple  and  durable. 

The  Venturi  Meter  (Fig.  118)  is  so  called  from  Venturi, 
who  first  pointed  out  the  relation  between  the  pressures  and 
velocities  of  flow  in  converging  and  diverging  tubes. 


FIG.  118. 

As  shown  by  the  longitudinal  section,  Fig.  1 19,  this  meter 
consists  of  two  truncated  cones  joined  at  the  smallest  sections 
by  a  short  throat-piece.  At  A  and  B  there  are  air-chambers 
with  holes  for  the  insertion  of  piezometers,  by  which  the  fluid 
pressure  may  be  measured.  By  Art.  5,  Chap.  I,  the  theoretical 
-quantity  Q  of  flow  through  the  throat  at  A  is 


a,a.y 


so  8 


WATER-METERS. 


# ! ,  a2  being  the  sectional  areas  at  A  and  B,  respectively,  and 
H2  —  HI  the  difference  of  head  in  the  piezometers,  or  the 
"head  on  Venturi, "  as  it  is  called. 


FIG.  rig. 

Introducing  a  coefficient  of  discharge  C,  the  actual  delivery 
through  A  is 


Q  =  C 


An  elaborate  series  of  experiments  by  Herschel  gave  C 
values  varying  between  .94  and  1.04,  but  the  great  majority  of 
the  values  lay  between  .96  and  .99. 


FIG.  120. — Schonheyder's  Positive 
Meter. 


FIG.  121. — The  Universal 
Meter. 


The  piezometers  may  be  connected  with  a  recorder,  and 
thus  a  continuous  register  of  the  quantity  of  water  passing 
through  the  meter  may  be  obtained  at  any  convenient  position 
within  a  radius  of  1000  ft.  This  distance  may  be  extended  to 
several  miles  by  means  of  an  electric  device. 


WATER-METERS. 


209 


Other  meters  may  be  generally  classified  as  Piston  or 
Reciprocating  Meters  and  Inferential  or  Rotary  Meters.  They 
are  all  provided  with  recorders  which  register  the  delivery  with 
a  greater  or  less  degree  of  accuracy. 

The  piston  meter  (Fig.  120)  is  the  most  accurate  and  gives 
a  positive  measurement  of  the  actual  delivery  of  water  as 
recorded  by  the  strokes  of  the  piston  in  a  cylinder  which  is 
filled  from  each  end  alternately.  Thus  an  additional  advantage 


FIG.  122.— The  Buffalo  Meter. 


FIG.  123. — The  Union  Rotary 
Piston  Meter. 


possessed  by  a  water-engine   is  that  the  working  cylinder  will 
also  serve  as  a  meter. 

In  inferential  meters  a  drum  or  turbine  is  actuated  by  the 
force  of  the  current  passing  through  the  pipe,  but  it  often 
happens  that  when  the  flow  is  small  the  force  is  insufficient  to 
cause  the  turbine  to  revolve,  and  there  is  consequently  no 
register  of  the  corresponding  quantity  of  water  passing  through 
the  meter. 


210  EXAMPLES. 


EXAMPLES. 

(N.B.    Take^1  =  32  and  6£  gallons  =  i  cu.  ft.  unless  otherwise  specified.) 

1.  A  water-main  is  to  be  laid  with  a  virtual  slope  of  i  in  850,  and  is 
to  give  a  maximum  discharge  of  55  cubic  feet  per  second.     Determine 
the  requisite  diameterof  pipe  and  the  maximum  velocity,  taking/=.oo64. 

Ans.  3.679  ft.;  3.2888  ft.  per  sec. 

2.  Find  the  loss  of  head  due  to  friction  in  a  pipe  ;  diameter  of  pipe 
=  12  in.,  length  of  pipe  =  5280  ft.,  velocity  of  flow  =  3  ft.  per  second  ; 

f  =  .0064.     Also  find  the  discharge. 

Ans.  19.008  ft.;  2.3562  cu.  ft.  per  sec. 

3.  A  pipe  has  a  fall  of  10  ft.  per  mile  ;  it  is  10  miles  long  and  4  ft.  in 
diameter.     Find  the  discharge,  assuming/  =  .0064. 

Ans.  54.7  cu.  ft.  per  sec. 

4.  A  pipe  discharges  250  gallons  per  minute,  and  the  head  lost  in 
friction  is  3  ft.     Find  approximately  the  head  lost  when  the  discharge 
is  300  gallons  per  minute  ;  also  find  the  work  consumed   by  friction  in 
both  cases.  Ans.  4.32  ft.;  7500  ft.-lbs.;  12,960  ft.-lbs. 

5.  What  is  the  mean  hydraulic  depth  in  a  circular  pipe  when   the 

diameter    . 
water  rises  to  the  height —  above  the  centre  ? 

2  V2  10 

Ans,  —   x  diameter. 
33 

6.  A  12-inch  pipe  has  a  slope  of  12  feet  per  mile ;  find  the  discharge. 
(/  =  .005.)  Ans.  2.118  cu.  ft.  per.  sec. 

7.  The  mean  velocity  of  flow  in  a  24*in.  pipe  is  5  ft.  per  second;  find 
its  virtual  slope,  f  being  .0064.  Ans.  i  in  200. 

8.  Calculate  the  discharge  per  minute  from  a  24-5n.  pipe  of  4000  ft. 
length  under  a  head  of  80  ft.,  using  a  coefficient  suitable  for  a  clean  iron 
pipe.  Ans.  34.909  cu.  ft.  per  sec. 

9.  How  long  does  it  take  to  empty  a  dock  whose  depth  is  31  ft.  6 
ins.  and  which  has  a  horizontal  sectional  area  of  550,000  sq.  ft.,  through 
two  7-ft.  circular  pipes  50  ft.  long,  taking  into  account  resistance  at  en- 
trance ?  Ans.  214  min.  6  sec. 

10.  The  virtual  slope  of  a  pipe  is  i  in  700;  the  delivery  is  180  cubic 
feet  per  minute.     Find  the  diameter  and  velocity  of  flow. 

Ans.  i. 26  ft.;  2.401  ft.  per  sec. 

11.  Determine  the  diameter  of  a  clean  iron  pipe,  100  feet  in  length, 
which  is  to  deliver  .5  cu.  ft.  of  water  per  second  under  a  head  of  5  feet. 
Assume/ =  .006.  Ans.  .328  ft. 

12.  A  reservoir   of   10,000  sq.   ft.  area  and   100  ft.  deep  discharges 


t 
EXAMPLES.  211 

through  a  pipe  24  ins.  in  diam.  and  2000  ft.  in  length.  Find  the  velocity 
•of  flow.  What  should  the  diam.  be  in  order  that  the  reservoir  may  be 
emptied  in  two  hours?  (/=.oo64.)  Ans.  15.37  ft.  per  sec.;  4.0923  ft. 

13.  The  pressure  from  an  accumulator  at  the  entrance  of  a  df-in.  pipe 
L  ft.  long  is  looo  Ibs.  per  sq.  in.     If  A^  is  the  total  H.P.  available  at  the 

N  \*L 


inlet,  show  that  the  H.P.  absorbed  in  frictional  resistance  isf 

/being  —  -  =  .0081. 

IS  14.  The  delivery  at  the  end  of  a  3-inch  pipe  is  11.06  H.P.  The  total 
effective  head  at  the  entrance  to  pipe  is  896  feet.  The  loss  in  frictional 
resistance  is  21  per  cent.  Find  the  distance  to  which  the  energy  is 
transmitted.  Ans.  15,000  ft.,/ being  .0064. 

^15.  A  reservoir  has  a  superficial  area  of  12,000  ft.  and  a  depth  of  60 
ft.;  it  is  emptied  in  60  minutes  throuali/0«r  horizontal  circular  pipes, 
equal  in  diameter  and  50  ft.  long.  Fiftd  the  diameter.  (/  =  .0064.) 

Ans.  1.786  ft. 
Explain  how  the  total  head  is  made  up,  and  draw  the  plane  of  charge. 

16.  A  3-inch  pipe  is  very  gradually  reduced  to  ^  inch.     If  the  pres- 
sure-head in  the  pipe  is  40  ft.,  find  the  greatest  velocity  with  which  the 
water  can  flow  through.  Ans.  1.4  ft.  per  sec. 

17.  Water  flows  through  a  24-inch  pipe  5000  yards  in'length.   At  1000 
yards  it  yields  up  300  cubic  feet  per  minute  to  a  branch.     At  2800  yards 
it  yields  up  400  cubic  feet  per  minute  to  a  second  branch.     At  4000 
yards  it  yields  up  600  cubic  feet  per  minute  to  a  third  branch.     The  de- 
livery at  the  end  is  500  cubic  feet  per  minute.     Find  the  head  absorbed 
by  friction.     (/  =  .0075.)  Ans.  177.801  ft. 

/^  18.  Find  the  H.P.  required  to  raise  550  gallons  per  minute  to  a  height 
of  60  feet,  through  a  pipe  100  feet  in  length  and  6  in.  in  diameter,  the 
coefficient  of  friction  being  .0064.  Ans.  10.74. 

19.  What  head  of  water  is  required  for  a  5-in.  pipe,  150  ft.  in  length, 
to  carry  off  25  cu.  ft.  of  water  per  minute  ?  Ans.  1.56223  ft. 

What  head  will  be  required  if  the  pipe  contains  two  rectangular 
knees?  Ans.  1.84918  ft. 

20.  Determine  the  delivery  of  a  2-in.  pipe,  48  ft.  long,  under  a  5-ft. 
head, /being  .005.  Ans.  .1449  cu.  ft.  per  sec. 

What  will  be  the  delivery  if  the  pipe  has  5  small  curves  of  90°  cur- 
vature, the  ratio  of  the  radius  of  the  pipe  to  that  of  the  curves  being 
1:2?  Ans.  .1381  cu.  ft.  per  sec. 

21.  The  curved  buckets  of  a  turbine  form  channels   12  in.  long,  2  in. 
wide,  and  2  in.  deep  ;  the  mean  radius  of  curvature  of  the  axis  is  8  in.; 
the  water  flows  along  the  channel  with  a  velocity  of  50  ft.  per  minute. 
What  is  the  head  lost  through  curvature?  Ans.  .00138  ft. 

22.  Find  the  power  transmitted  by  water  flowing  at  80  ft.  per  sec.  in 
a  36-inch  pipe,  the  metal  being  i|  inches  thick  and  the  allowable  stress 


212  EXAMPLES. 

2800  Ibs.  per  square  inch.     If  the  pipe  is  i£  miles  in  length,  find  the  loss 
of  power.  Ans.  576  H.P.;  720.2  ft.-lbs. 

23.  Find  the  diameter  of  a  pipe  \  mile  long  to  deliver  1500  gallons  of 
water  per  minute  with  a  loss  of  20  feet  of  head,     (f  =  .005.) 

Ans.  1.0135  ft- 

24.  Water  is  to  be  raised  20  ft.  through  a  3o-ft.  pipe  of  6  in.  diameter. 
Find  the  velocity  of  flow,  assuming  that  10  per  cent  of  additional  power 
is  required  to  overcome  friction,  and  that/  =  .0075. 

Ans.  8.44  ft.  per  sec. 

25.  In  a  pipe  3280  ft.  in  length  and  delivering  6750  gallons  per  min., 
the  loss  of  head  in  friction  is  83  ft.    Taking/  —  .0064,  find  the  diameter. 

Ans.  1.527  ft. 

26.  Calculate,  by  Thrupp's  formula,  the  flow  through  a  4-in.  rough 
wrought-iron  pipe  having  a  fall  of  33  feet  per  mile. 

Ans.  .1426  cu.  ft.  per  sec. 

27.  A  clean   6-in.  pipe  has  a'  virtual  slope  of    i    per  400.     Taking 
/  =  .005,  find  the  velocity  of  steady  flow,  the  discharge,  and  the  energy 
absorbed  in  frictional  resistance  in  1000  feet. 

Ans.  2  ft.  per  sec.;  ££  cu.  ft.  per  sec.;  6iT\3^  ft.-lbs. 

28.  A  6-in.  pipe,  500  ft.  long,  discharges  into  a  3-in.  pipe,  also  500  ft. 
long.     The  effective  head  between  the  inlet  and  outlet  is  10  feet.     Find 
the  discharge,  taking  /  =  .0064,  and  making  allowance  for  the  resistance 
at  the  inlet.  Ans.  .1703  cu    ft.  per  sec. 

29.  How  far  can  100  H.P.  be  transmitted  by  a  3|-in.  pipe  with  a  loss 
of  head  not  exceeding  25  per  cent  under  an  effective  head  of  750  Ibs.  per 
square  inch  ?  Ans.  5426.3  ft. 

30.  A  pipe  2000  ft.  long  and   2  ft.  in  diameter  discharges  at  the  rate 
of  16  ft.  per  second.     Find  the  increase  in  the   discharge  if  for  the  last 
looo  ft.  a  second  pipe  of  same  size  be  laid  by  the  side  of  the  first  and 
connected  with  it  so  that  the  water  may  flow  equally  well  along  either 
pipe.  Ans.  7.24  cu.  ft  per  sec. 

31.  A  pipe  of  length  /and  radius  r  gives  a  discharge  Q.     How  will 
the  discharge  be   affected  (i)   by   doubling  the    radius  for   the  whole 
length  ;  (2)  by  doubling  the  radius  for  half  the  length  ;  (3)  by  dividing  it 

into  three  sections  of  equal  length,  of  which  the  radii  are  r,  —  ,  and  —  , 

2  4 

respectively?     (/=.  coefficient  of  friction.) 

Ans.   i.  New  discharge  =  4 


64/ 


64/7U 
33//;  ' 


3.      «  «  =,    Q(  _  9r  +  I2//       \* 

^524.7  1  2r  +  4228/7;  ' 

32.  A  24-in.  pipe  2000  ft.  long  gives  a  discharge  of  Q  cubic  feet  of 
water  per  minute.  Determine  the  change  in  Q  by  the  substitution  for 
the  foregoing  of  either  of  the  following  systems  :  (i)  two  lengths,  each 


EXAMPLES.  213 

of  looo  ft.,  whose  diameters  are  24  ins.  and  48  ins.  respectively  ;  (2)  four 
lengths,  each  of  500  ft.,  whose  diameters  are  24  ins.,  18  ins.,  16  ins.,  and 
24*ins. 

Draw  the  "  plane  of  charge  "  in  each  case. 

Ans.  (i)  Discharge  is  increased  33.2  per   cent  taking  loss  at 

change  of  section  into  account; 
Discharge  is  increased  35.7  per  cent  disregarding  loss 

at  change  of  section. 
(2)  Discharge   is   diminished   45    per   cent   disregarding 

losses  at  change  of  section. 

33.  Q  is  the  discharge  from  a  pipe  of  length  /and  radius  r\  examine 
the  effect  upon  Q  of  increasing  r  to  nr  for  a  length  ml  oi  the  pipe. 

f  3.2/7  I* 


2 

f+  — (I   -m+-\  + 


Ans.   New  discharge  =  Q  , ^—  _   (^a  _ 

r   I1."*1"1"**;   ' 


34.  A  5-in.   pipe,  300  ft.  long,  discharges  into  a  3-in.  pipe,  200  ft. 
long,  the  total  fall  being  5  feet.     Find  the  quantity  of  flow  in  gallons 
per  hour.  Ans.  4080. 

35.  A  main,  1000  ft.  long  and  with  a  fall  of  5  ft.,  discharges  into  two 
branches,  the  one  750  ft.  long  with  a  fall  of  3  ft.,  ihe  other  250  ft   long 
with   a  fall  of  i   ft.     The  longer  branch  passes  twice  as  much  water  as 
the  other  and  the  total  delivery  is  47}  cu.  ft  per  minute.     The  velocity 
of  flow  in  the  main  is  2|  ft.  per  second      Find  the  diameters  of  the  main 
and  branches.     (/ =  .0064.)  Ans.   63245  ft.  ;  .51  ft.  ;  .36  ft. 

36.  The  water  in  a  12-in.  main.  800  ft.  long,  flows  at  the  rate  of  i  ft. 
per  second  and  one  third  of  the  water  ;s  discharged  into  a  branch  200  ft. 
long  with  a  fall   of    i  in   40,  while   the   remainder   passes  into  a  6oo-ft. 
branch  with  a  fall  of  i  in  60.     The  effective  head  between  the  inlet  and 
outlet  of  the  main  is  i\  ft.     Find  the  total  discharge  and  the  diameters 
of  the  branches,  taking  f  —  .0064,  and  making  allowance  for  loss  at  inlet 
but  disregarding  losses  at  the  Junction. 

Ans.  94!  cu.  ft.  per  sec. ,  .27  ft. ;  .39  ft. 

37.  If  a  pipe  whose  diameter  is  8  ins.  suddenly  enlarges  to  one  whose 
diameter   is   12  ins.,  find  the  power  required  to  force  1000  gallons  per 
minute  through  the  enlargement,  and  draw  to  scale  the  plane  of  charge. 

Ans.    Energy  expended  =  .1377  H.P. 

38.  1000  gallons   per  minute  are   forced  through  a  system  of  pipes 
AB,  BC,  CD.  of  which  the  lengths  are  100  ft.,  50  ft.,  and  120  ft.,  and  the 
radii  6  ins.,  3  ins.,  and  4  ins.,  respectively.     Draw  to  scale  the  plane  of 
charge. 

Ans.  Loss  in  friction  from  A  to  B  —    14.744  ft.;  ioss  at  B  =  14.56  ft. 
"     "        "  "      B  to  C-  235.9      "  :     4<     "  C  —    8.819" 

'•     "         "  "      C  toD=  134.36     " 


214  EXAMPLES. 

39.  A  pipe  4  ins.  in  diameter  suddenly   contracts   to   one  3   ins.    iir 
diameter ;  find    the  power  necessary   to   force  250  gallons  per  minute 
through  the  sudden  contraction.  Ans.  1.23997  H.P. 

40.  Water  flows  from  a  3-in.  pipe  through  a  i^-in.  orifice  in  a  dia- 
phragm into  a  2-in.  pipe.     What  head  is  required  if  the  delivery  is  to  be 
8  cu.  ft.  of  water  per  minute?  Ans.  2.826  ft. 

41.  500  gallons  of  water  per  minute  are  forced  through  a  continuous 
line  of  pipes  AB,  BCt  CD,  of  which  the  radii  are  3  ins.,  4  ins.,  2  ins.,  and 
the  lengths  100  ft.,  150  ft.,  and  80  ft.,  respectively.     Find  the  total  loss 
of  head  (a)  due  to  the  sudden  changes  of  form  at  B  and  C,  (£)  due  to 
friction.     Find  (c)  the  diameter  of  an  equivalent   uniform   pipe  of  the 
same  total  length. 

Ans.  (a)  .1378  ft.;   1.152  ft. 

(ff)  3.688  ft.  in  AB\  1.313  ft.  in  BC\  22.393  ft-  m  c^- 
(c)  .4212  ft. 

42.  AB,  BC,  CD  is  a  system  of  three  pipes  of  which  the  lengths  are 
1000  ft.,  50  ft.,  and  800  ft.,  and  the  diameters  24  ins.,  12  ins.,  and  24  ins., 
respectively;  the  water  flows  from  CD  through  a  i-in.  orifice  in  a  thin 
diaphragm,  and  the  velocity  of  flow  in  AB  is  2  ft.  per  second.     Draw 
the   plane   of    charge    and  find    the   mechanical   effect   of   the    efflux, 
/being  .0064. 

Ans.  Loss  at  C  —  -^  ft.  ;  at  B  =  -jfa  ft.;  in  friction  from  A  to 
B  —  .8  ft. ;  from  B  to  C  —  1.28  ft. ;  from  C  to  D  =  .64  ft.  ;  energy 
of  jet  =  14,81  if  H.P. 

43.  1000  gallons  per  minute  flows  through  a  sudden  contraction  from 
12   ins.  to  8  ins.  at  A,  then  through  a  sudden  enlargement  from  8  ins.  to 
12  ins.  at  B,  the  intermediate  pipe  AB  being  100  ft.  long.     Draw  the 
plane  of  charge, /being  .0064. 

Ans.  Loss  at  A  =  .288  ft. ;  at  B  —  .281  ft.  ;  in  friction  from  A 
to  B  =  3.499  ft. 

44.  Water  flows  from  one  tube   into  another  of  twice  the  diameter  ; 
the  velocity  in  the  latter  is  10  ft.     Find  the  head  corresponding  to  the 
resistance.  Ans.  14.0625  ft. 

45.  A  2-in.  pipe  A  suddenly  enlarges  to  a  3-in.  pipe  B,  the  quantity 
of  water  flowing  through  being  100  gallons  per  minute.     Find  the  loss 
of  head  and  the  difference  of  pressure  in  the  pipes  (i)  when  the  flow  is 
from  A  to  B ;  (2)  when  the  flow  is  from  B  to  A,  Cc  being  .66. 

Ans.  (i)  Loss  of  head  =    8.639  m- 

Gain  of  pressure-head  =  13.83     " 

(2)  Loss  of  head  =    7.428  " 

Diminution  of  pressure-head  =  29.88     " 

46.  A  3-in.  horizontal  pipe  rapidly  contracts  to  a   i-in.  mouthpiece, 
whence  the  water  emerges  into  the  air,  the  discharge  being  660  Ibs.  per 
minute.     Find  the  pressure  in  the  3-in.  main. 


EXAMPLES.  2  1  5 

If  the  3-in.  pipe  is  200  ft.  in  length  and  receives  water  from  an  open 
ta'ak,  find  the  height  of  the  tank,/  being  .005. 

Ans.  1003.5  Ibs.  per  sq.  ft.  ;   19.92  ft. 

47.  A   horizontal   pipe   4   ins.    in    diameter    suddenly   enlarges  to  a 
diameter  of  6  ins.  ;  find  the  force  required  to  cause  a  flow  of  300  gallons 
of  water  per  minute  through  the  sudden  enlargement. 

Ans.  .06  H.P. 

48.  1000  gallons  per  minute  is  to  be  forced  through  a  system  of 
pipes  AB,  BC,  CD,  of  which  the  lengths  are  100  ft.,  50  ft.,  120  ft.,  and 
the  radii  4  ins.,  6  ins.,  and  3  ins.,  respectively.     What  must  be  diameter 
of  equivalent  uniform  pipe  ?     Draw  the  plane  of  charge,  /being  .0064. 

Ans.  Diameter  =  3.4  ins.  ; 

loss  in  friction  from  A  to  B  =  1  1  1.96    ft.;  loss  at  B  =    4.499  ft.; 
"     "        "  "     B  to  C  =      7.372    "      "     "    C=  14.56    " 

"     "         "  "      CtQjD=  566.17      " 

49.  Find  the  H.P.  required  to  pump  1,000,000  gallons  of  water  per 
day  of  24  hours  to  a  height  of  300  ft.  through  a  line  of  straight   piping 
3000  ft.  long,  the  diameter  of  the  pipe  being  8  ins.  for  the  first  loco  ft., 
6  ins.  for  the  second,  and  4  ins.  for  the  third,  allowance  being  made  for 
the  loss  at  inlet  and  the  losses  at  abrupt  changes  of  section  ;  also  4  is  to 
be  taken  as  the  coefficient  of  resistance  for  pump-valves.     (At  changes 
of  section  ce  =  .64.)     What  is  the  diameter  of   an  equivalent    uniform 
pipe?  (/=  .0064.)  Ans.   196;  diam.  =  .403  ft.,  or  say  5  ins. 

50.  In  a  given  length  /  of  a  circular  pipe  whose  inner  radius  is  r  and 
thickness  e,  a  column  of  water  flowing  with   a  velocity  v  is  suddenly 
checked  by  the  shutting  off  of  cocks,  etc.     Show  that 


in  which  //  =  head  due  to  the  velocity  v,  E  =  coefficient  of  elasticity, 
E\  =  coefficient  of  compressibility  of  water,  A  =  extension  of  pipe  cir- 
cumference due  to  E. 

51.  The  water  surface  in  one  reservoir  is  500  ft.  above  datum,  and  is 
100  ft.  above  the  surface  of  the  water  in  a  second  reservoir  20,000  ft. 
away,  and  connected  with  the  first  by  an  i8-in.  main.     Find  the  delivery 
per  second,  taking  into  account  the  loss  of  head  at  the  entrance. 

Ans.  7.64  cu.  ft.  per  sec.,  /being  .0064. 

52.  Determine  the  discharge  from  a  pipe  of  12  in.  radius  and  3280  ft. 
in  length  which  connects  two  reservoirs  having  a  difference  of  level  of 
128  ft.     Take  into  account  resistance  at  entrance.     Draw  the  plane  of 
charge.     (/  =  .005.)  Ans.  48.571  cu.  ft.  per  sec. 

53.  Determine  the  diameter  of  a  clean  iron  pipe  5000  ft.  in  length 
which  connects  two  reservoirs  having  a  total  head  of  40  ft.  and  dis- 
charges into  the  lower  at  the  rate  of  20  cu.  ft.  per  second.     Draw  to 
scale  the  line  <.f  charge.     (/=  .005.)  Ans.  1.9219  ft. 


216  EXAMPLES, 

54.  The  difference  of  level  between  the  two  reservoirs  is  100  ft.,  and 
they  are  connected  by  a  pipe  10,000  ft.  long.     Find  the  diameter  of  the 
pipe  so  as  to  give  a  discharge  of  2000  cubic  feet  per  minute  (a)  by  Darcy's 
formula,  (b}  assuming/  =  .0064.     (Allow  for  loss  of  head  at  entrance.) 

Ans.  (a)  2.256  ft.  if  a=  .0001622  ;  (b)  2.360  ft. 

55.  Two  reservoirs  are  connected  by  a  12-inch   pipe  ij  miles  long. 
For  the  first  500  yards  it  has  a  slope  of  i  in  30,  for  the  next  half  mile  a 
slope  of  i  in  100,  and  for  the  remainder  of  its  length  it  is  level.     The 
head  of  water  over  the  inlet  is  55  ft.  and  that  over  the  outlet  is  15  ft. 
Determine  the  discharge  in  gallons  per  minute.     (Take/  =  .0064.) 

Ans.  1950.66. 

56.  Two  reservoirs  are  connected  by  a  6-in.  pipe  in  three  sections, 
each  section  being  three  quarters  of  a  mile  in  length.      The  head  over 
the  inlet  is  20  ft.,  that  over  the  outlet  9  ft.   The  virtual  slope  of  the  first 
section  is  i  in  50,  of  the  second  i  in  100,  and  the  third  section  is  level. 
Find  the  velocity  of  flow,  and  the  delivery,/ being  .005. 

Ans.  4.5  ft.  per  sec.;  332  gallons  per  minute. 

57.  A  pipe  5  miles  long,  of  uniform  diameter  equal  to  12  in.,  conveys 
water  from  a  reservoir  in  which  the  water  stands  at  a  height  of  300  tt. 
above  Trinity  high-water  mark,  to  a  reservoir  in  which  the  water  stands 
at  a  height  of  150  ft.  above  the  same  datum.    To  what  height  will  water 
rise   in  a  supply-pipe  taken  one  mile  from  the  lower  end  ?     For  what 
pressure  would  you  design  the  main  at  this  point,  if  it  lies  20  ft.  above 
the  level  of  the  lower  reservoir?     (/  =  .0064.) 

Ans.  179.755  ft.;   19.13  Ibs.  per  sq.  in. 

58.  A  clean  6-in.  pipe,  1000  ft.  long,  has  four  sharp  knees,  viz.,  one 
of  60°,  two  of  90°,  and  one  of  120°.     Find  the  head  wasted  at  the  knees 
and  in  the  straight  pipe,  the  flow  being  at  the  rate  of  150  gallons  per 
minute.  Ans.   .2734  ft.;  3.0237  ft. 

59.  A  6-in.  pipe,  4000  feet  in  length",  leads  from  a  reservoir  A  to  a. 
point  O,  at  which  it  divides  into  two  6-inch  branches,  each  4000  feet  in 
length,  the  one  leading  to  a  reservoir  B,  the  other  to  a  reservoir  C. 
The  surface  of  the  water  in  A  is  100  feet  above  that  in  B  and  200  feet 
above  that  in   C.     Find  the  velocities  of  flow    in  the  three    branches, 
/being  .0064.  Ans.  v\  =  7.89  ft.  per  second  —  z/3;  v^  —  o. 

60.  A  pipe  24  in.  in  diameter  and  2000  ft.  long  leads  from  a  reservoir 
in  which  the  level  of  the  water  is  400  ft.  above  datum  to  a  point  B,  at 
which  it  divides  into  two  branches,  viz.,  a  i2-in.  pipe  JSC\  1000  ft.  long, 
leading  to  a  reservoir  in  which  the  surface  of  the  water  is  250  feet  above 
datum,  and  a  branch  BD,  1500  ft.  long,  leading  to  a  reservoir  in  which 
the  surface  of  the  water  is  50  ft.  above  datum.     Determine  the  diameter 
of  BD  when  the  free  surface-level  at  B  is  (a)  300  ft.;  (b}  250  ft.,  and  (c) 
200  above  datum.  Ans.  (a)  1.454  ft.;  (b)  1.783  ft.;  (c)  2.096  ft. 

61.  Two  reservoirs  A  and  B  are  connected  by  a  line  of  piping  MON, 
2000  ft.  in  length.     From  the  middle  point  O  of  this  pipe  a  branch  OP, 
1000  ft.  in  length,  leads  to  a  reservoir  C.     The  reservoirs  A  and  B  are 


EXAMPLES.  2 1 7 

200  feet  and  100  feet,  respectively,  above  the  level  of  C.  The  deliveries 
in  MO,  OP,  ON,  in  cubic  feet  per  second,  are  V^.V-*.  and  *  respectively. 
Ffnd  (a)  the  velocities  of  flow  in  MO,  OP,  ON\  (b)  the  radii  of  these 
lengths;  (c)  the  height  of  the  free  surface-level  at  O  above  C,/being 
.0064.  Ans.  (a)  11.121  ft.  per  sec.  in  MO ;  10.158  ft.  per  sec.  in  OP; 

14.145  ft.  per  sec.  in  ON. 
(b)  .5  ft.;  .41831  ft.;  .26588  ft.         (c)   150.5  ft.,  very  nearly. 

62.  Find  the  amount  of  water  in  gallons  per  day  which  will  be  de- 
livered by  a  24-inch  cast-iron  pipe,  15,000  ft.  in  total  length,  when  the 
water  surface  at  the  outlet  is  87^  ft.  below  the  water  surface  at  the  inlet, 
taking/  =  .001  and  allowing  for  resistance  at  inlet. 

If  the  water,  instead  of  flowing  into  a  reservoir,  is  made  to  drive  a 
reaction  turbine,  what  must  be  the  velocity  of  flow  in  the  pipe  to  give  a 
max.  speed?  What  will  be  the  H.P.  of  the  turbine  if  its  efficiency  is  .84? 
A  third  reservoir  is  connected  with  the  system  by  means  of  a  24-111. 
cast-iron  pipe,  7500  ft.  long,  joined  to  the  main  at  the  middle  point. 
The  water  surface  of  this  intermediate  reservoir  is  50  ft.  above  that  of 
the  lowest  reservoir.  Discuss  the  distribution. 

Ans.  22,628, 57if;  7.7  ft.  per  sec. ;  12.63  H.P. ;  2  =  73.68  or 
51.32  ft.;  v\  —  7.76  or  12.42  ft.  per  sec.;  z/a  =  10.05  or  2-373  ft.  per 
sec.;  7/1  =  17.73  or  !4-S  ft.  per  sec. 

63.  The  water-levels  in  two  reservoirs  A  and  B  are,  respectively,  300 
ft.  and  200  ft.  above   that  in  C.     The  reservoir  A   supplies  3  cu.  ft.  of 
water,  of  which  2  cu.  ft.  go  to  B  and  i  cu.  ft.  goes  to  C.    A  pipe  2500  ft. 
long  leads  from  A  to  a  junction  at  O,  from  which  two  branches,  each  2500 
ft.  in  length,  lead,  the  one  to  B  and  the  other  to  C.     Assuming  that  the 
cost  of  laying  a  pipe  in  place  is  proportional  to  the  diam.  and  that  this 
cost  is  to  be  a  minimum,  find  the  pressure  head  at  O  and  the  diams.  of 
the  pipes. 

Ans.   164  ft.;  diam.  of  AO  =  .66  ft.,  of  OB  —  63  ft.,  of  OC  =  .4  ft. 

64.  An  engine  pumps  a  volume  of  Q  cubic  feet  of  water  per  second 
through    a  hose    i  ft.  in   length,  and  d  feet  in  diameter,  having  at  the 
end   a   nozzle  D  feet  in   diameter.     Find  the  pumping  H.P.  and  apply 
your  result  to  the  determination  of  the  H.P.  of  an  engine  which  is  to 
pump  30  cu.  ft.  of  water  per  minute  through  a  i-in.  nozzle  at  the  end  of 
a  3-in.  hose  400  ft.  in  length  (/=  .00625).     Also  find  the  force  required 
to  hold  the  nozzle.  Ans.  ii¥3^  H.P.  ;  89$!  Ibs. 

65.  A  fire-engine  pumps  water  through  a  4oo-ft.  length  of  2^-in.  bore 
at  the  rate  of   12  ft.  per  second,  and  discharges  through  a  i-in.  nozzle. 
Find  the  pressure  in  the  hose,  and  the  pumping  H.P.     Also  find  the 
force  required  to  hold  the  nozzle,     (f  ==  .00125.) 

Ans.  .6702  Ibs.  per  ft.;  5.0916;  59.95  Ibs. 

66.  The  conduit-pipe  for  a  fountain  is  250  ft.  long  and  2  in.  in  diam- 
eter ;  the  coefficient  of  resistance  for  the  mouthpiece  is  .32 ;  the  entrance 
orifice  is  sufficiently  rounded,  and  the  bends  have  sufficiently  long  radii 


2 1 8  EXAMPLES. 

of  curvature  to  allow  of  the  corresponding  coefficient  of  resistance  being- 
disregarded.      How  high  will  a  ^-in.  jet  rise  under  a  head  of  30  ft.  ? 

Ans.  20.4  ft. 

67.  Water  surface  of  a  reservoir  is  300  ft.  above  datum,  and  a  4-in. 
pipe  600  ft.  long  leads  from  reservoir  to  a  point  200  ft.  above  datum. 
Find  the  height  .to  which  the  water  would  rise  (a)  if  end  of  pipe  is  open 
to  atmosphere,  (b)  if  it  terminates  in  a  i-in.  nozzle.     In  latter  case  find 
longitudinal  force  on  nozzle.       Ans.  (a)  2f  ft.  ;  (ff)  87.52  ft.;  59.693  Ibs. 

68.  The  surface  of  the  water  in  a  tank  is  388  ft.  above  datum  and  is 
connected  by  a  4-in.    pipe   200   ft.   long  with  a  turbine   146  ft.  above 
datum.     Determine  the  velocity  of  the  water  in  the  pipe  at  which  the 
power  obtained  from  the  turbine  will  be  a  maximum.     Assuming  the 
efficiency  of  the  turbine  to  be   85   per  cent,    determine   the  power,  f 
being  .005.  Ans.  19.928  ft.  per  sec.  :  27.11075  H.P. 

69.  A  pipe  12  ins.  in  diameter  and  900  ft.  long  is  used  as  an  inverted 
siphon  to  cross  a  valley.     Water  is  lead  to  it  and  away  from  it  by  an 
aqueduct  of  rectangular  section  3  ft.  broad  and  running  full  to  a  depth 
of  2  ft.  with  an  inclination  of  i  in  1000.     What  should  be  the  difference 
of  level   between   the  end  of  one   aqueduct  and   the  beginning  of  the 
other, /being  .0064  for  the  pipe,  and  .008  for  the  aqueduct  ? 

Ans.  14.39. 

70.  Water  flows  through  a  pipe  20  ft.  long  with  a  velocity  of   10  ft. 
per  second.     If  the  flow  is  stopped  in  TV  second  and  if  retardation  during 
the  stoppage  is  uniform,  find  the    increase  in    the    pressure  produced. 
(g  =  32  and  the  density  of  the  water  =  62.5  Ibs.  per  cu.  ft.) 

Ans.  62^  cu.  ft.  of  water. 

71.  An  hydraulic  motor  is  driven  by  means  of  an  accumulator  giving 
750  Ibs.  per  square  inch.     The  supply-pipe  is  900  ft.  long  and  4  ins.   in 
diameter.     Find  the  maximum  power  attainable,  and  velocity  in  pipe. 
(/  =  .0075.)  Ans^  242.4  H.P.  ;  21.203  ft-  per  sec. 

72.  A  2-in.  hose  conveys  2  gallons  of  water  per  second.     Find   the 
longitudinal  tension  in  the  hose.  Ans.  9.18  Ibs. 

73.  Find  the  pumping  H.P.  to  deliver  i  cu.  ft.  of  water  per  second 
through  a  i-in.  nozzle  at  end  of  a  3-in.  hose  200  ft.  long,/ being  .016. 

Ans.  97.335  H.P. 

74.  The  surface  of  the  water  in  a  tank  is  286  ft.  above  datum.     The 
tank  is  connected  by  a  4-in.  pipe    500  ft.    long  with  a  36-in.    cylinder 
170  ft.  above  datum.     Find  (a)  the  velocity  of  flow  in  the  pipe  for  which 
the  available  power  will  be  a  maximum;  (£)  the  power.     If  the  piston 
moves  at  the  rate  of  i  ft.  per  minute,  find  (c)  the  pressure  on  the  piston. 
Also  find  the  height  to  which  the  water  would  rise  if  (d)  the  cylinder 
end  of  the  pipe  were  open  to  the  atmosphere  and  if  (e)  the  pipe  termi- 
nated in  a  nozzle  i    in.   in  diameter,  neglecting  the  frictional  resistance 
of  the  nozzle.     Finally,  find  (/)  the  power  required  to  hold  the  nozzle. 
(Coeff.  of  friction  =  .005.)         Ans.  (a)    8.93    ft.  per  sec.;  (£)  6.85   H.P.  ; 
(c)  22.8  tons  per  sq.  ft. ;  (d)  3.74  ft.  ;  (e)  103.8  ft.  ;  (/)  70.8  Ibs. 


EXAMPLES.  219 

75.  A    3-in.   hose,    400  ft.    in   length,  terminates  in  a  f-in.   nozzle; 
Water  enters  the  hose  under  ahead  of  297!-  ft.     Find  the  velocity  of 
efflux,  the  height  to  which  the  issuing  jet  will  rise,  the  pressure-head  at 
the  nozzle  inlet,  and   the   force   required    to  hold  the   hose,  f  being 
.00625.         Ans.  128  ft.  per  sec. ;  256  ft.  ;   18,437^  Ibs.  per  sq.  ft.  ;  98!  Ibs. 

76.  A  reducer,    10  ft.  long,  conveys  400  gallons  of  water  per  min- 
ute, and  its  diameter  diminishes  from   12  ins.  to  6  ins.;  find  the  total 
loss  of  head  due  to  friction.  Ans.  .05529. 

77.  A  reservoir  is  to  be  supplied  with  water  at  the  rate  of    11,000 
gallons  per  minute,  through  a  vertical  pipe  30  ft.  high;  find  the  mini- 
mum diameter  of  pipe  consistent  with  economy.     Cost  of  pipe  per  foot 
=  §d,  d  being  the  diameter  ;  cost  of  pumping  =  I   cent  per  H.P.   per 
hour  ;  original  cost  of  engine  per  H.P.  =  $100.00;  add  10  per  cent  for 
depreciation.     Engine  works   12  hours  per  day  for  300  days  in  the  year, 
/being  .0064.  Ans.  4.375  ft. 

78.  A  city  is  supplied  with  water  by  means  of  an  aqueduct  of  rect- 
angular section,  24  ft.  wide,  running  4  ft.  deep,  and  sloping  i  in  2400. 
One-fourth  of  the  supply  is  pumped  into  a  reservoir  through  a  pipe  3000 
ft.  long,  rising  25  ft.  in  the  first  1500  ft.,  and  75  ft.  in  the  second  1500  ft. 
The  pumping  is  effected  by  an  engine  burning  2|  Ibs.  of  coal  per  H.P. 
per  hour,  and  working  constantly  through  the  year.     A  percentage  is 
to  be  allowed  for  repairs  and  maintenance ;  the  cost  of  the  coal  per  ton 
of  2000  Ibs.  is  $4;  the  prime  cost  of  the  engine  is  $100  per  H.P.;  the 
efficiency  of  the  engine  is  f ;  the  coefficient  of  pipe  friction  is  .0064,  the 
cost   of   the    piping    is   $30  per  ton.     Determine  the  most  economical 
diameter  of  pipe,  and  the  H.P.  of  the  engine,  /being  .0064  for  the  pipe 
and  .08  for  the  channel.  Ans.  4.84  ft.;  456.455  H.P. 

79.  A  vessel  with  500  sq.  ft.  of  surface  experiences  a  resistance  of  150 
Ibs.   per  sq.  ft.  when  steaming  at  5  knots.     How  much   H.P.  will  be 
absorbed  in  frictional  resistance  by  a  vessel  with  10,000  sq.  ft.  of  surface 
steaming  at  18  knots?  Ans.  2140^. 

80.  The  performances  of  two  similarly  designed  ships  are  to  be  com- 
pared.    The  one,  with  a  length  of  300  ft.  and  a  displacement  of  8000 
tons,  is  to  steam  at  20  knots.     What  should  be  the  length  and  displace- 
ment of  the  other,  which  is  to  steam  at  21  knots?     Compare  also  the 
I.H.P.s.  Ans.  33of  ft.;   10,720  tons;   1.34. 

81.  From  a  central  junction  four  mains,  each  10,000  ft.  long,  lead  to 
four  reservoirs,  A,  B,  C,  D,  the  water-levels  in  A,  B,  C  being  600,  400, 
and  200  ft.,  respectively,  above  that  in  D.     If  the  diameter  of  each  main 
is  12  ins.,  find  (a)  the  effective  head  at  the  junction  and  the  velocities  of 
flow.     If  the  velocity  in  each  main  is  5  ft.  per  sec.,  find  (b)  the  effective 
head  at  the  junction  and  the  diameters  of  the  mains. 

Ans.  (a)  300  ft,;  8.66  */s  in  highest  and  lowest  mains;  5  f/s  in 

intermediate  mains. 

(b)  300  ft.;  4  ins.  for  highest  and  lowest  mains  ;   12  ins. 
for  intermediate  mains. 


CHAPTER    III. 
FLOW   OF   WATER    IN   OPEN   CHANNELS. 

I.  Channel-flow  Assumptions. — A  transverse  section  of 
the  water  flowing  in  an  open  channel  may  be  supposed  to 
consist  of  an  infinite  number  of  elementary  areas  representing 
the  sectional  areas  of  fluid  filaments  or  stream-lines.  The 
velocities  of  these  stream-lines  are  very  different  at  different 
points  of  the  same  transverse  section,  and  the  distribution  of 
the  pressure  is  also  of  a  complicated  character.  Generally 
speaking,  the  side  and  bed  of  a  channel  exert  the  greatest 
retarding  influence  on  the  flow,  and  therefore  along  these  sur- 
faces are  to  be  found  the  stream-lines  of  minimum  velocity. 
The  stream-lines  of  maximum  velocity  are  those  farthest 
removed  from  retarding  influences.  If  the  stream-line  veloci- 
ties for  any  given  section  are  plotted,  a  series  of  equal  velocity- 
curves  may  be  obtained.  In  a  channel  of  symmetrical  section 


FIG.  124. 

the  depth  of  the  stream-line  of  maximum  velocity  below  the 
water-surface  is  less  than  one  fourth  of  the  depth  of  the  water, 
while  the  mean  velocity-curve  cuts  the  central  vertical  line  at 

220 


CHANNEL-FLOW  ASSUMPTIONS.  221 

a  point  below  the  surface  about  three  fourths  of  the  depth  of 
tfre  water. 

In  the  ordinary  theory  of  flow  in  open  channels  the  varia- 
tion of  velocity  from  point  to  point  in  a  transverse  section  is 
disregarded,  and  it  is  assumed  that  all  the  stream-lines  are 
sensibly  parallel  and  move  normally  to  the  section  with  a 
common  velocity  equal  to  the  mean  velocity  of  the  stream. 
With  this  assumption,  it  also  necessarily  follows  that  the  dis- 
tribution of  pressure  over  the  section  is  in  accordance  with  the 
hydrostatic  law. 

Again,  it  is  assumed  that  the  laws  of  fluid  friction  already 
enunciated  are  applicable  to  the  flow  of  water  in  open  channels. 
Thus  the  resistance  to  flow  is  proportional  to  some  function 
of  the  velocity  (F(v)\  to  the  area  (S)  of  the  wetted  surface,  is 
independent  of  the  pressure,  and  may  be  expressed  by  the  term 
5 .  F(v)-  An  obvious  error  in  this  assumption  is  that  v  is  the 
mean  velocity  of  the  stream  and  not  the  velocity  of  the  stream- 
lines along  the  bed  and  sides  of  the  channel.  In  practice, 
however,  the  errors  in  the  formulae  based  upon  these  imperfect 
hypotheses  are  largely  neutralized  by  giving  suitable  values  to 
the  coefficient  of  friction  (/). 

When  a  constant  volume  (Q)  of  water  feeds  a  channel  of 
given  form,  the  water  assumes  a  definite  depth,  a  permanent 
regime  is  said  to  be  established  and  the  flow  is  steady.  If  the 
transverse  sectional  area  (A)  is  also  constant,  then,  since 
Q  —  vA,  the  velocity  v  is  constant  from  section  to  section  and 
the  flow  is  said  to  be  uniform.  Usually  the  sectional  area  A 
is  variable  and  therefore  the  velocity  v  also  varies,  so  that  the 
motion  is  steady  with  a  varying  velocity.  Any  convenient 
short  stretch  of  a  channel,  free  from  obstructions,  may  be 
selected  and  treated,  without  error  of  practical  importance,  as 
being  of  a  uniform  sectional  area  equal  to  that  of  the  mean 
section  for  the  whole  length  under  consideration. 

2.  Steady  Flow  in  Channels  of  Constant  Section  (A). — 
The  flow  is  evidently  uniform;  and  since  A  is  constant,  the 


STEADY  FLOW  IN   CHANNELS. 

depth  of  the  water  is   also   constant,  so  that  the  water-surface 
^^ft  ..  T  is  parallel  to  the  channel-bed. 

;  \G 

Consider     a     portion     of    the 
^  «feg^   stream,    of  length    /,    between 
the     two    transverse    sections 


^*§^^^^^^S^::§M^^ 


Let    /   be    the    inclination 
of   the   bed   (or  water-surface) 
FlG>  I25'  to  the  horizon. 

Let  P  be   the   length   of  the  wetted  perimeter   of  a  cross- 
section. 

Then,  since  the  motion  is  uniform,  the  external  forces  act- 
ing   upon    the   mass   between   aa   and   bb    in    the   direction   of 
motion  must  be  in  equilibrium. 
These  forces  are : 

(1)  The  component  of  the  weight  of  the  mass,  viz., 

li'Al  sin  /'  =  iv  A  It  =  ivAl—j  =  wAk, 

h  being  the  fall  of  lexrel  in  the  length  /. 

NOTE. — When  /  is  small,  as  is  usually  the  case  in  streams, 

h 

—  =  tan  *  =  sin  t  =  /,  approximately. 

(2)  The  pressures  upon  the  areas  aa  and  bb,  which  evidently 
neutralize  each  other. 

(3)  The  frictional  resistance    developed  by  the  sides  and 
bed,  viz., 

Hence 

wAh  -  PlF(v)  =  o, 
or 

F(v)  _Ah  _ 
10       =  ~Pl  ~  m*' 

m  being  the  hydraulic  mean  depth. 


STEADY  FLO  IV  IN  CHANNELS.  223 

It  now  remains  to  determine  the  form  of  the  function  F(v). 
In  ordinary  English  practice  it  is  usual  to  take 


W  2g* 

/"being  the  coefficient  of  friction.      Then 


f—  —  mi. 
J 


/2g 

-  A/  ~£~  ^mi  =  c  V mi, 


or 


C  being  a  coefficient  whose  value  depends  upon  the  roughness 
.of  the  channel  surface  and  upon  the  form  of  its  transverse 
section. 

The  total  head  H  in  a  stream  is  made  up  of  two  parts,  the 
one  being  utilized  in  producing  the  velocity  of  flow  and  the 
other  being  absorbed  in  frictional  resistance.  Thus 


2g     '     111     W 

In  long  channels  and  in  rivers  in  which  the  slope  of  the  bed 

?£ 

does  not  exceed   3   ft.  per  mile  the  term  -  -  is  very  small  as 

2<£~ 

compared  with  -         -  and  may  be   disregarded  without  sensi- 
ble error.      In  this  case 


m    w 


Ex.  i.  A  channel  of  regular  trapezoidal  section,  with  banks  sloping 
at  30°  to  the  vertical,  has  a  bottom  width  of  8  ft.,  and  a  width  of  16  ft.  at 
the  free  surface.  It  conveys  288  cu.  ft.  of  water  per  sec.,  and  the  fall  is 
i  in  2000.  Find  the  mean  depth,  the  mean  velocity  of  flow,  and  the 
coefficients  /"and  c. 


224  EXAMPLES. 

Depth  of  waterway  =  4  tan  60°  =  4  ^3  ft. 

A  =  ^-(8  +  16)44/3"=  48V3~sq.  ft. 
P  =  8  +  2   x  4  4/3  sec  30°  =  24  ft. 

Therefore  the  mean  depth  =  — -  =  24/3  =  3.464  ft. 

ooo 

The  mean  velocity  of  flow  =   — =  =,3^3"=  3.464  ft.  per  sec. 

45V  3 

Hence 


'      J  2000  2000 

Therefore 

/  =  .009237     and     c  —  82.63. 

Ex.  2.  How  much  water  is  conveyed  away  by  a  horizontal  trench 
10  ft.  wide,  the  depth  of  the  water  at  entrance  being  5  ft.,  and  the  sur- 
face falling  i  ft.  in  2400  feet?  (Take/  =  .008.) 

Area  at  upper  end  =  50  sq.  ft. ;  at  lower  end  =  40  sq.  ft. ; 

the  mean  area  =  —(40  +  50)  =  45  sq.  ft. 
Therefore,  if  Q  cu.  ft.  are  conveyed, 

—  =  velocity  at  upper  end  ;  —  =  that  at  lower  end  ; 
and 

mean  velocity  =  — . 
45 

The  wetted  perimeter  =  20  ft.  at  ^upper  end  ;  =  1 8  ft.  at  low  er  end  ; 
and 

mean  wetted  perimeter  =  -  (20  +  18)  =  19  ft. 

Thus  the  hydraulic  mean  depth  m  —  — . 
Hence 

!  +  (2.Y-L  =  (-2.V  —  -  •°°8  x  2400 /gyj^ 

\So)   64       \4o)    64  f|  \45y     64' 

and 

Q  =  389  cu.  ft.  per  sec. 

3.  Retarding  Effect  of  Air,  etc. — The  retarding  effect  of 
the   air  upon  the   free   surface  of  a  river  or  of  the  water  in   a 
canal  or  in  any  channel  has  not  yet  been  accurately  determined. 
It  may  be  assumed  that  the  resistance  per  unit  of  free  surface 


RETARDING   EFFECT  OF  AIR,  ETC.  225 

•due  to  the  air  is  about  one  tenth  of  the  resistance  due  to  similar 
unks  at  the  bottom  and  sides  of  smooth  channels.  Thus  if 
JC  is  the  width  of  the  free  surface  in  a  smooth  channel,  the 

wetted  perimeter  becomes  P  -| . 

In  general,  the  wetted  perimeter  may  be  expressed  in  the 
form  P  -f-  — ,  ft  being  10  for  smooth  channels  and  greater  than 

10  for  rough  channels.     The  value  of/?  is  evidently  diminished 
by  opposing  winds  and  increased  by  following  winds. 
Again,  in  the  formula 

.      F(v) 
mi  =  -  — , 
W 

m  f=  jy]  and  i  l=  -j\  are  similarly  related  in  the  deter- 
mination of  7>,  the  mean  velocity  of  flow.  If  v  is  constant,  the 
product  mi  must  also  be  constant,  so  that  if  m  increases  i  must 
diminish,  and  vice  versa.  Thus  in  a  very  flat  country  the 
flow  may  be  maintained  by  making  m  sufficiently  large,  while, 
again,  if  the  channel-bed  is  steep  m  is  small. 

The  erosion  caused  by  a  watercourse  increases  with  the 
rapidity  of  flow.  At  the  same  time  the  sectional  area  (A)  of 
the  waterway  also  increases,  so  that  the  velocity  of  flow  v 
diminishes.  Thus  there  is  a  tendency  to  approximate  to  a 
<*  permanent  regime  "  when  the  resistance  to  erosion  balances 
the  tendency  to  scour. 

Hence,  throughout  any  long  stretch  of  a  river  passing 
through  a  specific  soil,  the  mean  velocity  of  flow  will  be  very 
nearly  constant  if  the  amount  of  flow  (Q)  does  not  vary. 
Generally  speaking,  the  volume  conveyed  by  a  river  increases 
from  source  to  mouth  on  account  of  the  additions  received  from 
tributaries,  etc.  Since  Q  increases,  A  must  also  increase;  and 
if  mi  or  v  is  to  remain  constant,  i  must  diminish.  It  is  to  be 
observed  that  the  surface  slopes  of  large  rivers  diminish 
gradually  from  source  to  mouth. 


226  RETARDING  EFFECT  OF  AIR,  ETC. 

For  a  given  discharge  (Q)  the  mean  depth  (m)  diminishes, 
as  i  increases,  and,  as  the  cost  of  constructing  a  canal  is 
approximately  proportional  to  the  mean  depth,  it  is  advisable 
to  give  the  bed  as  large  a  slope  as  possible.  But  the  velocity 
of  flow  (v)  also  increases  with  i,  and  the  slope  must  therefore 
not  exceed  that  for  which  v  would  be  so  great  as  to  cause  the 
erosion  of  the  banks.  On  the  other  hand,  v  must  not  be  so 
small  as  to  allow  of  the  growth  of  aquatic  plants  or  of  the 
deposition  of  sand,  gravel,  and  other  detritus,  which  would 
soon  obstruct  the  waterway  and  add  a  considerable  item  to  the 
cost  of  maintenance.  Between  these  extreme  limits  the  slope 
may  be  varied  in  any  required  manner,  the  controlling  influ- 
ences being  the  configuration  of  the  ground  and  the  nature  of 
the  soil  through  which  the  canal  passes.  In  every  case  a 
careful  determination  should  be  made  of  the  best  combination 
of  the  three  elements  7',  /,  and  A  which  would  give  a  specified 
discharge.  In  France  the  canal  beds  have  slopes  varying 
frorn  I J  to  20  in  10,000,  and  the  magnitudes  of  both  v  and  t 
may  be  considerable  when  the  canal  passes  through  rock  or 
through  a  well-compacted  material  capable  of  resisting. erosion. 
According  to  Belgrand  the  value  of  v  for  water  carrying  fine 
particles  of  loam  should  exceed  I  ft.  (.25  m.)  per  secondhand 
should  not  be  less  than  2  ft.  (.5  m.)  per  second  if  the  waters 
are  laden  with  coarse  particles v  of  loam  or  sand.  In  clear 
water,  the  growth  of  weeds,  etc.,  which  would  seriously  inter- 
fere with  the  flow,  is  prevented  if  the  velocity  of  flow  is  from 
2  to  3  ft.  (.5  m.  to  .8  m.)  per  second. 

The  slope  of  an  aqueduct,  in  which  no  trouble  is  to  be 
anticipated  from  plant-growth,  may  be  as  small  as  3  in  10,000, 
and  may  even  fall  to  I  in  10,000  when  the  waters  are  excep- 
tionally clear,  as  in  the  case  of  the  aqueducts  on  the  Dhuis  and 
Vanne.  On  the  other  hand,  the  slope  should  rarely,  if  ever, 
exceed  12  in  10,000,  and  as  a  general  rule  the  slope  should  be 
less  than  10  in  10,000.  The  ordinary  channel  formula,  viz., 
•u  =  c  Viniy  is  applicable  to  the  flow  in  a  conduit,  so  long  as  the 


RETARDING   EFFECT  OF  AIR,  ETC. 


22J 


conduit  does  not  run  full,  and  since  v  is  proportional  to  Vm  it 
is.  a  maximum  for  some  definite  depth  of  water.  When  the 
water  fills  the  conduit,  the  formula  for  channel-flow  ought  to> 
change  suddenly  so  as  to  agree  with  that  for  pipe-flow,  and  in 
this  respect  the  theory  is  therefore  imperfect.  The  mean 
velocity  of  flow  in  a  conduit  should  not  be  less  than  about 
2  ft.  (0.5  m.)  per  second,  and  may  be  as  great  as  5f  ft., 
(1.5  m.)  per  second.  High  velocities  enable  the  waters  to> 
carry  off  floating  debris  and  sand  particles.  There  should  be 
no  sudden  changes  of  slope  or  of  section,  as  they  favor  the 
formation  of  eddies  and  the  deposition  of  detritus. 

The  following  table  of  slopes  and  mean  velocities  is  taker* 
from  the  article  by  Daries  in  the  Encycl.  Sc.  des  Aide- 
Memoire : 


Slope  in 
10,000. 

Mean  Velocity  per 
Second. 

Nature  of  Canal  Sides. 

Feet 

Metres. 

Craponne  Canal  
Marseille  "  
Carpentras  "  

Saint-Martory  Canal  . 
Verdon  Canal 

IO 

3  to  7 
2  to  4 

5.4 

i-5  to  2 

2 

2  tO   I 
2 
1.2 

3 

5 
2-5 

2.1 

Alluvial  soil 
Earth  and  rock 
Vegetable      soil,     fis- 
sured limestone 
Clay 
Calcareous  rocks 
Clay,   pudding-stone^,, 
rock 
Alluvial  soil 
Calcareous  rocks 
Earth 
Millstone-grit  mason- 
ry   with    a    ff-inch 
(=   .02    m.)    facing 
in  cement 
Millstone-grit  mason-  - 
ry    with    a   ff-inch 
(=  .02  m.)  facing  in 
cement 
Rubble  masonry  with 
a  |f-  in.  (=  .oism.) 
facing  in  cement 

Rubble  masonry  with 
a||-in.  (=  .015  m.> 
facing  in  cement  .. 

3-3 
2.53  to  6.56 

1.64 

2-5 
1.64 

1.  21 
1.67 
I.3I 

1.18 
3-3 

2-95 

2.62 
2.36 

I 

.77  tO   2 

•  5 
.76 
•5 

•37 
•51 
•4 
.36 

i 

•9 

.8 
.72 

Neste  '  '  

Beaucaire  Canal.  .... 

Dhuis  Aqueduct  

Naples  Aqueduct  .... 

Montpellier  Aqueduct 
Croton  Aqueduct  

228^       ON   THE  FORM  OF   THE  SECTION  OF  A   CHANNEL. 

4.  On  the  Form  of  the  Section  of  a  Channel. — The  funda- 
mental formulae  governing  the  form  of  the  transverse  section 
of  a  channel  are 

Q  =  Av 
and 

7,2  __  #mi  __  #    t. 

Therefore,  also, 

Pi*  =  c*Qi. 

For  channels  of  the  same  slope 

r2  oc  ;//. 

Take  v^  —  am,  a  being  some  constant. 

Then,  if  dv  is  a  small  change  in  the  velocity  corresponding 
to  a  small  change  dm  in  the  hydraulic  mean  depth, 

2v  .  dv  =  a  .  dm, 
and  therefore 

dv       dm 

i'   ~     2m 

Thus  the  hydraulic   mean  depth  must  be  changed  20  per 
cent  to  produce  a  change  of  10  per  cent  in  the  velocity. 
Again, 

Q  a  Pi*. 

But  P  increases  with  Q,  and  therefore  Q  increases  more 
rapidly  than  ?'3.  For  example,  an  increase  in  the  velocity  of 
less  than  3-J-  per  cent  will  cause  an  increase  of  10  per  cent  in 
the  discharge. 

For  channels  giving  the  same  discharge 

/V3  oc  i. 

For  a  given  volume  of  water  there  must  be  a  sensible 
change  in  the  slope  to  produce  an  appreciable  change  in  the 


ON   THE  FORM   OF   THE  SECTION   OF  A   CHANNEL. 


229 


velocity  of  flow,  although,  generally  speaking,  the  wetted  pe- 
rimeter (P)  diminishes  or  increases  as  /  increases  or  dimin- 
ishes, and  thus  v.  and  therefore  -^,  increases  or  diminishes 

yi 

more  rapidly  than  i.  An  increase  of  10  per  cent  in  the  ve- 
locity causes  a  diminution  of  about  4  per  cent  in  the  sectional 
area  of  the  waterway. 

For  channels  of  the  same  slope  and  giving  the  same  dis- 

A* 
charge  Pv3  and  also  -rj   are  constant.      A  further  condition  is 

required  before  the  sectional  area  can  be  determined. 

PROBLEM  I.  A  canal  of  rectangular  section  and  of  width 
x  is  to  convey  water  of  depth  y 
with  the  condition  that  either  the 
sectional  area  (A)  of  the  waterway 
is  to  be  a  constant  quantity  or  the 
wetted  perimeter  (P)  is  to  be  a 
minimum.  It  is  proposed  to  find 
the  relation  between  x  and  y  so 


w- 

;A 

£(/////Y//////////////Sj//////////////////< 

1 

FIG.  126. 


that    (a)    the    velocity  of  flow    may  be  a  maximum,   (b)  the 
quantity  of  flow  may  be  a  maximum. 


and 


If  v  is  a  maximum., 


If  Q  is  a  maximum, 


P  .  dA  -  A  .  dP 


3A2  .  PdA  —  A*  .  dP 


230         ON   THE  FORM  OF   THE  SECTION  OF  A   CHANNEL. 

In  each  case,  if  dA  =  o,  i.e.,  if  the  area  is  constant,  then 

and  if  dP  —  o,  i.e.,  if  the  wetted  perimeter  is  a  minimum,  then 

dA  =  o. 

Thus  the  same  results  are  obtained  for  the  problem  in  its 
different  conditions. 
Now 

A  =  xy     and     P  =  x  -f-  2y. 
Therefore 

dA  =  y  .  dx  -f-  x  .  dy  =  o, 

and 

dP  =  dx  -f-  2  .  dy  =  o. 
Hence 

x 

Therefore,  also, 


A       x 


XI 


and 


2  V  2 

A  suitable  value  for  c  corresponding  to  the  slope  i  or  to  the 
value  of  m  f  =  — )  can  be  obtained  from  the  Tables  of  Bazin, 
Kutter,  or  Manning  at  the  end  of  the  chapter. 


ON   THE  FORM  OF  THE  SECTION  OF  A   CHANNEL.        231 

PROBLEM   II.      The   section   is   usually  in   the   form   of  a 
quadrilateral,   the  non-parallel  sides  sloping  at    an  angle,    #, 


FIG.  127. 

depending    upon    the    nature   of  the   soil   through   which  the 
channel  passes. 

For  example,  in  a  canal 

with  retaining  walls  6  =  63°  36', 

with  stiff  earthen  sides,  faced,        0  =  45°, 
with  stiff  earthen  sides,  unfaced,    0  =.  33°  41', 
with  sides  in  light  or  sandy  soils  0  =  26°  34'. 

In  such  a  channel  let  x  be  the  bottom  width  and  y  the 
depth  of  the  water.  Then,  the  remaining  conditions  being 
the  same  as  those  in  Problem  I,  it  again  follows  that 

dA  —  o     and     dP  =  o. 
But 

A  —  y(x  -f- y  cot  ff)     and     P  =  x  -f-  2y  cosec  0. 
First.      If  0  is  given, 

dA  =  o  =  y  .  dx  +  (x  +  2 y  cot 
and 

dP  =  o  ==  "dx  -f-  2  cosec  0  .  dy. 

Therefore 

2y  cosec  0  =  x  -^  2y  cot  0, 


232         CW   THE  FORM  OF  THE  SECTION   OF  A   CHANNEL. 


or 


or 


x  sin  6  =  27(1  ~~  cos  #), 


tan  -  =  —  . 

2          2 


The  section  may  be  easily  sketched,  as  in   Figs.   128  and 


129. 


A  C  B 

FIG.  128. 


From  the  middle  point  C  of  AB,  the  bottom  width,  draw 
CF  at  right  angles  to  AB  and  equal  in   length  to  the  depth  of" 
the  water.      Then 

AB  0 

CF=2t™2> 

0  being  the  given  slope  of  the  sides. 

With  F  as  centre  and  FC  as  radius  describe  a  circle. 
From  the  points  A  and  B  draw  tangents  to  touch  this  circle  at 
D  and  E.  FA  evidently  bisects  the  angle  CAD.  Therefore 

CAD  CF        CF  B 

tan  -      -  =  tan  CAF  —  -^  —  ——^  —  cot  -. 
2  AC       ^AB  2 

Hence    n  —  CAD  =  6,    and    AD,    BE    have    the    slope 
required. 
Again, 


2  —  cos  B 
sin  B     ' 


ON   THE  FORM  OF  THE  SECTION  OF  A  CHANNEL.         233. 
or 


VA  sin  8 
2  -  COS  B 

and 

1  —  cos  0    , 

/>=  27-2^3-+  2,  cosec* 

2  —  COS  #          2y2 

=  2'-ifirF-  =7- 

Therefore 

A       y 


and 


the  coefficient  c  being  obtained  from  the  tables. 

The  following  Table  gives  the  best  relative  values,  per  unit 
cf  area,  of  x,  y,  m,  and  P,  corresponding  to  specified  values 
of  0,  and  the  actual  values  may  be  obtained  by  multiplying 
those  of  the  Table  by  VA: 

9  x  y  m  P 

90°  I.4I4         .707         -3535         2.828 

_6o°_  .877  .760  .380  2.633 

45°  .613  .740  .370  2.706 

4°°  .525  .722  .361  2.772 

36°  52'  .471  .707  -3535  2.828 

35°  -439  .697  .3485  2.870 

30°  .„  .336  .664  .332  3.012 

^26°  34'  .300  .636  .318  3.144 

The  above  values  cannot  always  be  exactly  adopted  in 
actual  practice.  The  character  of  the  soil,  the  importance  of 
preventing  excessive  filtration,  and  the  difficulties  of  construe- 


234         ON   THE  FORM  OF   THE  SECTION  OF  A   CHANNEL 

tion  and  maintenance,  often  render  it  necessary  to  insure  that 
the  depth  of  the  water  shall  not  exceed  a  certain  limit,  say  8 
to  12  ft.  (2   m.  to  3   m.).      In  France  the  depth  of  irrigation- 
canals  is  between  4  and  6^  ft.  (1.2  m.  and  2m.). 
Second.     If  the  bottom  width  x  is  fixed,  then 

dA  =  o  =  (x  -\-  2y  cot  &)dy  —  y2-  cosec2  &  .  dO 
and 

dP  =  o  =  2  cosec  6  .  dy  —  2y   .   2  ^  .  dO. 

Hence 

x  -\-  2y  cot  0  y 


2  cosec  0  2  cos  0' 

or 

x  sin  0  cos  0  —  —  j(2  cos2  0  — 
or 


and  therefore 


sin  20  a 

x —  —  y  cos  20, 


tan  (?r  —  20)  =  --  tan  20  =  — . 


It  may  be  observed  that  as  the  width  (x)  of  the  bottom 
increases,  0  also  increases. 

If  the  width  is  nil,  then  tan  20  =  oo  and  0  =  45°,  so  that 
the  triangular  section  of  minimum  perimeter  is  a  semi-square. 

Third.     If  the  depth  y  is  fixed,  then 

dA  =  o  =  ydx  —  y2  cosec2  0  .  d& 
and 


dP  =  O  =  dx  -  2J/  .  dO. 

J  sin2  0 


Therefore 


or 

cos  0  =  i     and      0  =  60°. 


ON   THE  FORM  OF   THE  SECTION  OF  A   CHANNEL.         235 

PROBLEM  III.  To  find  the  proper  sectional  form  of  a 
channel  of  bottom  width  2a  so  that  the  mean  velocity  of  flow 
may  be  constant  for  all  depths  of  water. 

Let  x,  j/,  Fig.  1 30,  be  the  co-ordinates  of  any  point  P  in 
the  profile  referred  to  the  middle  point  O  of  AB,  the  bottom 
width,  as  origin,  and  let  s  be  the  length  of  AP. 


FIG.  130. 

Since  v  is  to  be  constant,  m  must  also  be  constant,    and 
therefore 

A    fy-dx 

-p:  =  =  a  const.  —  m, 

P         s  +  a 

which  may  be  written 

/ y  .dx  =  m(s  -f-  «). 
Differentiating, 

y  .  dx  —  m  .  ds  —  m(dx* 
and  therefore 

dx  __         dy 


Integrating, 

x 

-  —  log^  (j/  -)-  i/j/2  —  m2)  -\-  Cj 

c  being  a  constant  of  integration. 


But  y  =  a  when  x  —  o,  and  .  • .  o  =  \oge(a  -f-  Va2 — 


236          ON    THE  FORM   OF   THE  SECTION   CF  A   CHANNEL. 
Hence,  too, 


Adding  together  the  last  two  equations, 

*        m*  -*- 
2y  —  be™  +  -e   "', 


or 


y  — 


which  is  the  equation  to  the  required  profile,  and  is  a  curve 
which  belongs  to  the  class  of  catenaries  and  which  evidently 
flattens  out  very  rapidly. 

If  the  bottom  width  is  such  that 


a  =  m  —  b, 


the  equation  becomes 


and  the  profile  is  a  true  catenary  of  parameter  m,  with  its  axis 
coincident  with  the  bottom  and  vits  directrix  coincident  with 
the  vertical  at  the  middle  of  the  section. 


FIG.  131. 


PROBLEM  IV.  A  channel  of  given  slope  has  a  given  sur- 
face width  AC,  vertical  sides  AB  (=  j^)  and  CD  (=  j/2)  of 
given  depths,  and  a  curved  bed  BD  (=  L)  of  given  length. 


ON   THE  FORM  OF   THE  SECTfOM  OF  A   CHANNEL         237 

The  amount  and  velocity  of  flow  in  the  channel  will  be  a 
maximum  when  the  form  of  the  bed  BD  is  a  circular  arc. 
This  can  be  easily  proved  as  follows : 

Since  the  slope  is  constant,  v  oc  Vm  oc 

But  P  (—  L  -\-  yl  +  y^  is  a  constant  quantity,  and  there- 
fore v  and  also  Q  will  be  a.  maximum  when  A  is  a  maximum. 

Hence,  too,  the  area  between  the  chord  BD  and  the  curve 
must  be  a  maximum,  and  therefore  the  curve  must  be  a  circular 
arc.  The  proof  of  this  by  the  Calculus  of  Variations  is  as 
follows : 

Take  O  in  CA  produced  as  the  origin,  OC  as  the  axis  of 
,#•,  and  the  vertical  through  O  as  the  axis  of/.  Then 

A  =    I  ** ydx  is  to  be  a  maximum. 

t/JCl 


dy 
is  a  given  quantity,  OA  being  =  x^ ,  OC  =  *2 ,  and  -- 


Let  V '  =  y  -\-  a  V /i  -|-  /2,  a  being  some  constant. 
Then 

/    *  V  .  dx  is  to  be  a  maximum, 

tJxl 

and  therefore 

dV 

v  =  *Tp  +  c- 

that  is, 


and  thus 


238         CW   THE  FORM  OF  THE  SECTION  OF  A   CHANNEL. 
Therefore 

dx        i  c,  —  v 


Integrating, 


P        Va*  —  (^  —  y 


=  W-  (c.-y?, 


the  equation  to  a  circle  of  radius  a. 

Hence  the  profile  BD  is  a  circular  arc. 
The  maximum  depth  of  the  channel  is  cl  —  a. 
The  constants  cl ,  c9 ,  a  can  be  found  from  the  three  condi- 
tions that  the  arc   is  of  given  length  and  has  to  pass  through 
the  two  fixed  points  B  and  D. 

PROBLEM  V.  The  Semicircular  Channel. — Theoretically > 
the  best  form  of  channel  for  a  given 
waterway  is  one  in  which  the  bed  is 
a  circular  arc  (Prob.  IV),  as  the 
wetted  perimeter  is  then  a  minimum 
and  the  mean  depth  (or  radius)  a 
FIG.  132.  maximum. 

In  the  semicircular  channel,  Fig.   132,  let  the  free  surface 
subtend  an  angle  0  at  the  centre. 
Then 


A  =     (e  _  sin 


)(,  -5S. 

I  /3 

V  ^ 


and 


r  being  the  radius. 
Therefore 


A       r 

m  =  —  =  - 


sin  6 


ON   THE  FORM  OF   THE  SECTION  OF  A   CHANNEL. 

<22 

Hence,  since  mi  =  by*  =  b~, 

A 


If  the  channel  runs  full,  0  —  TT,  and  then 


As  a  first  approximation  it  may  be  assumed  that 

for  small  channel  sections  with  cement  faces  ......  b  =  .00022 

"  channels  of  mean  dimensions  with  smooth  faces  b  =  .00017 
<4  channels  of  large  dimensions  ................  b  =  .0001  1 

In  metric  measure  these  coefficients  become  .0004,  .0003, 
and  .0002,  respectively. 

Miscellaneous  Problems.  —  The  bed  of  the  aqueduct  at 
Naples  is  semi-elliptic,  but  beds  in  the  form  of  a  semi-ellipse, 
a  cycloid,  a  parabola,  or  an  hyperbola,  would  only  be  adopted 
under  very  exceptional  conditions,  as  when  a  curved  profile  is 
required  with  a  limited  depth.  The  waterway  and  the  wetted 
perimeter  can,  of  course,  be  approximately  calculated  from  the 
known  properties  of  these  curves. 

For  the  semi-elliptic  section,  if  a  and  b  are  the  semi-major 
and  minor  axes, 


ab 

A   =   7t~, 


and 


fwhere         t/=  i.     1.0025,    1.01,     1.0226,     1.0404,     1.0635,     1.0922,     1.1267.    I-1677,     1.2155;} 
1  when  ~^-6  =  °i         -i.          -2-  -3,  -4,  -5,  -6,  .7.  -8,  -9-       J 

For  the  cycloidal  section,  if  r  is  the  radius  of  the  generating 
circle, 

A  =     7tr*     and     P  =  Sr. 


240  AQUEDUCTS. 

Therefore 


and  the  flow  equation  becomes 


If  the  water-line  is  at  y2^4,  defined  by  the  angle  #  which 
the  radius  OA  of  the  generating  circle  makes  with  the  vertical, 
then 


-  2r2(i  —  cos  0)(7T  —  #  +  sin  61) 


=      ;r  _  0  +  2  sin  0  +  27i  cos  0  -  20  cos  0  + 


and 


8 
P  =  Srcos  -. 


5.  Aqueducts.  —  The  aqueduct  of  the  ancients  was  of 
rectangular  section  and  was  sometimes  of  very  large  dimen- 
sions as  compared  with  the  volume  of  water  to  be  conveyed. 
Although  in  modern  times  there  are  examples  of  rectangular 
sections,  it  is  now  more  usual  to  make  them  circular,  egg- 
shaped,  square  with  a  diagonal  vertical,  or  trapezoidal.. 
Aqueducts  are  also  constructed  of  forms  which  are  combina- 
tions of  the  circle  and  egg-shaped,  or  of  the  trapezoid  and  circle. 
When  a  mean  volume  of  water  is  to  be  conveyed  and  when 
provision  has  to  be  made  for  a  definite  height,  as,  for  example, 
for  a  man  standing  upright,  preference  is  given  to  the  egg- 
shaped  aqueduct. 

In  the  sections  shown  by  Figs.  133  to  137  it  will  be 
observed  that  a  rise  of  the  water-line  near  the  top  causes  an 


AQUEDUCTS. 


241 


appreciable  increase  in  the  wetted  perimeter,  while  there  is  no 
proportional  increase  in  the  waterway.  Thus  the  mean  depth 
(m)  and  therefore  also  the  mean  velocity  (v)  of  flow  continually 
diminish.  The  a  priori  conclusion  may  be  drawn  that  the 
discharge  (Q)  is  not  a  maximum  when  the  pipe  runs  full,  but 
when  the  water-line  is  some  distance  below  the  top.  The 


FIG.   133.  FIG.   134.  FIG.  135.  FIG.   136.  FIG.   137. 

differential  equation  defining  this  position  may  be  easily  found 
as  follows  (Prob.  i,  p.  229): 


Therefore 


A^i, 

~pi>' 


Since  Q  is  to  be  a  maximum, 


Therefore 


.dA-A*.dP 


or 


is  the  equation  required. 


242  AQUEDUCTS. 

If  the  velocity  of  flow  is  to  be  a  maximum, 

dv  =  o, 


and  therefore 


IA\       P  .dA  -  A.dP 
dm  =  o  =  a\-^l  =  -         —Fir, —     — , 


or 

P  .  dA-  A  .  dP  =  o. 

Ex.  I.      Circular  Section. — Let  the  wetted  perimeter  sub- 
tend an  angle   B  at   the  centre.       Then 

r  r* .  dB 

-— id— sin#)  and  dA=—    —  (i—  cos#); 

2  V  2      V 

P  =  rB      and     dP  =  r  .  dB. 
Hence  for  a  maximum  discharge 


2tf_3^Cos#+sin0  =  o. 

0  —  308°  is  the  value  of  B  satisfying  this  equation. 
For  a  maximum  velocity 


-dO  .  B(i  —  cos  0) (6  -  sin  0)  =  O, 

2  2 

or 

(9  =  tan  By 

and  0  =257°  27'  is  the  value  of  B  which  satisfies  this  equation. 

In  circular  aqueducts  the  angle   B  is  usually  about  240°, 
which  insures  a  certain  clear  space  above  the  water-line. 

Then,  also, 

P  =  4.189;-;    ^=2.528^;    ;//  =  .6r. 


AQUEDUCTS.  243 

EXAMPLE  2.     A   Square  Section  with  Vertical  Diagonal* 
-»i-Let  a  side  of  the  square  --  a, 
and    let   x  be   the   length   of  the 
portion  of  the  side  which  is  not 
wretted.      Then 


and 

dA  —  —  x  .  dx ; 

T-J 

and 

dP  =  -  2  .  dx.  FlG- 

Hence  for  a  maximum  discharge 


i          x%\ 
+  2(0*  —  —jdx  =  O, 


or 

$X*  —  I2«^r  -f-  20*  =  O. 
Therefore 


=  -(6-  ^26)=  .I8«, 


and  the  depth  below  the  apex  of  the  water-line 


=  -=  =  .1274^. 


For  a  maximum  velocity  of  flow 


or 

xz  —  ^.ax  -f-  2a*  =  p, 


244  AQUEDUCTS. 

and  therefore 

x  =  a(2  —  t/2*)  =  .58580, 

and  the  depth  of  the  water-line  below  the  apex 


=  —=  =  .41420. 
V2 


EXAMPLE  3.  Egg-shaped  Section. — This  form  of  aqueduct 
consists  essentially  of  three  parts,  a  lower  portion  bounded  by 
a  semicircle  of  radius  rl ,  an  upper  portion  bounded  by  a  cir- 


FIG.  140. 

cular  arc  of  lesser  radius  r2 ,  and  an  intermediate  portion 
bounded  by  circular  arcs  of  radius  r3 ,  which  meet  the  lower 
and  upper  arcs  tangentially. 

The  depth  of  the  intermediate  portion  is  defined  by  the 
angle  a  which  the  radius  O3O2  makes  with  the  horizontal,  and 
the  position  of  the  water-line  AA  is  defined  by  the  angle  & 
which  02A  makes  with  03O2  produced.  Then 


AQUEDUCTS.  245 

If  the  water-line  is  above  BB, 


and 


If  the  water-line  coincides  with  BB,  6  =  0,  and  then 

nr  2  r  2 

-4  =  ^  +  r32«  -  (rs  -  ^)(r3~  ra)  sin  «  +  ^  sin  2« 

and 

P  =  ^rx  -f-  2r3<*. 

If  ^  is  the  vertical  distance  between   Ol  and  the  highest 
point, 

z  =  r2  -\-  (r3  —  rt)  sin  <*. 
Also, 


If  the  water-line  CC  is  below  ^9j5,  let  0  be  the  angle  sub- 
tended at  <93  by  the  arc  .567,  and  let  6>2dT  =  x.     Then,  since 

is  now  <9, 


^r  sin  (6  —  0)  =  (r3  —  r2)  sin  0 
and 

x  cos  (#  —  a)  =  r3  cos  («  —  0)  —  (rs  —  rj), 

two  equations  giving  x  and  0  in  terms  of  6  and  the  radii. 

The  area  of  the  waterway  is  now  the  area  up  to  BB 
diminished  by  the  area  of  the  slice  between  BB  and  CC,  and 
this  area 

r2  x* 

=  —  sin  2a  +  r320  —  r3(r3  —  ra)  sin  0  -|  --  sin  2(#—  «). 
2  2 


246  AQUEDUCTS. 

Hence 

nr\        I  I      r2 

A=-£-  +  rf*-  (r3  -  ^)(rs  -  r2)  sin-ar  +  -±-  sin 


{r  2 
—  sin  2«  -j-  r320  —  r3(r3  —  r2)  sin  0  -[-  '—  sin  2(6— a] 


2 
—  r,)  sin  ^  —  r,  sin 


—  —  sin  2(^  —  n). 


and 


The  larger  diameter  is  usually  at  the  bottom  for  aqueducts, 
but  almost  invariably  at  the  top  for  sewers. 

The  discharge  for  sewers  may  be  calculated  by  Bazin's 
formula,  but  an  allowance  of  20  per  cent  should  be  made  in 
order  to  make  provision  for  deposits  and,  where  they  occur, 
for  water-pipes,  electric  conduits,  etc.  Care  should  also  be 
taken  that  the  section  is  sufficient  to  carry  away  the  water  from 
the  heaviest  rains  and  from  the  branch  drains  in  such  manner 
that  the  water  in  the  sewer  does  not  rise  above  a  certain  level. 

Assuming  that  the  time  of  flow  in  the  sewer  is  three  times 
that  of  the  rainfall  and  that  the  maximum  downfall  is  27.5 
gallons  (=125  litres)  per  second,  Belgrand  has  proposed  for 
the  discharge  of  the  Paris  sewers  the  formula 

5  X  .188  =  A  Vmi, 

S  being  the  drainage  area  in  acres. 

In  metric  measure,  5  being  the  drainage  area  in  hectares, 

5  X  .0239  =  A  Vmi. 

In  branch  drains  and  in  smaller  systems  the  influx  of  water 
is  much  more  rapid  and  the  time  of  flow  should  not  be  estimated 
at  more  than  twice  the  duration  of  the  rainfall. 


FORMUL/E  OF  PRONY,  EYTELH/EIN,  ETC.  247 

NOTE. — In  designing  sections  for  open  channels  or  aque- 
ducts, complicated  preliminary  calculations  may  be  generally 
avoided  by  employing  a  graphical  method.  Selecting  a  pro- 
visional section,  the  water  areas  and  wetted  perimeters  may  be 
obtained  for  different  depths  of  water  and  the  corresponding 
mean  depths  plotted  to  any  convenient  scale.  Repeating 
these  operations  for  different  sections,  the  mean-depth  curves 
will  quickly  indicate  the  best  section  to  be  adopted. 

6.  Formulae  of  Prony,  Eytelwein,  Beardmore  and  Tadini. 
— A  careful  study  of  Chezy's  experiment  on  the  Courpalet 
cut  (Orleans  canal)  and  of  twenty-three  experiments  made  by 
Dubuat  on  wooden  channels  of  small  section,  led  Prony,  in 
1804,  to  adopt  the  equation 

F(v) 

=  av  -4-  bv*1  =  mi. 

w 

in  which  —  =  22472.5  and  -r-  =  10607.02. 

About  the  year  1815,  Eytelwein,  taking  into  account  sixty 
additional  experiments  on  the  Rhine  and  Weser  by  Woltmann, 
Funk  and  Brunings,  proposed  slightly  different  values  for  a 
and  b,  viz., 


i  =  4l2II.H     and     ^=8975.43- 


The  expression  mi  has  the  same  value  with  Prony 's  as  with 
Eytelwein's  coefficients  when  the  velocity  is  about  72  ft.  per 
minute,  and  for  a  small  change  in  this  velocity  the  variation  in 
the  value  of  mi  is  also  small  and  of  little  practical  importance. 
For  other  velocities  the  value  of  mi  with  Prony 's  coefficients 
will  be  greater  or  less  than  the  value  with  Eytelwein's  coeffi- 
cients according  as  the  velocity  of  flow  is  greater  or  less  than 
72  ft.  per  minute. 

The  formula  with  Eytelwein's  coefficients  was  for  a  long 


248  FORMUL/E  OF  PRONY,  EYTELWEIN,  ETC. 

time  used  by  engineers,  and  was  preferred  as  giving  the  most 
reliable  results. 

For  values  of  v  exceeding  20  ft.  per  minute  the  term  av 
is  small  as  compared  with  bv*,  and  may  be  disregarded  without 
much  error,  This  formula  then  becomes 

bv*  =  mi, 
and  therefore,  according  to  Prony, 

v  =  — =  Vmi  =103  Vmiy 

V '  b 

and  according  to  Eytelwein, 

v  =  — =  Vmi  =  95  Vmi. 

Intermediate  between  these  is  Beardmore's  formula,  viz., 

v  =  100  \'mi. 
• 
Barre  de  St.  Venant  has  suggested  the  relation 

mi  —  .000136^" 

(or  mi  =  .OOO4^TT,  if  a  metre  is  the  unit). 

The  above  formulae,  now  obsolete,  involve  a  grave  error, 
as  it  is  assumed  that  the  resistance  due  to  the  roughness  of  the 
wetted  surface  is  a  constant  quantity.  Bazin's  experiments 
have  clearly  shown  that  the  resistance  may  vary  between  very 
wide  limits  depending  upon  the  nature  of  the  materials  and  soil 
which  form  the  bed  and  sides  of* the  channel.  For  a  deep  and 
wide  channel,  in  which  the  slope  of  the  bed  is  small,  approxi- 
mately accurate  results  are  given  by  Tadini's  formula, 

?/  =  91  Vmi 
(or  v  =    50  \/mi,  if  a  metre  is  the  unit). 


BAZIWS  FORMULAE. 


249 


7.  Bazin's  Formulae. — Between  1855  and  1859,  Darcy 
and  Bazin  carried  out  a  number  of  experiments  in  a  cut 
leading  from  the  Bourgogne  canal.  The  channel  sections  were 
of  different  forms  and  dimensions,  the  sides  were  faced  with 
wood,  cement,  hewn  ashlar,  bricks,  rubble  masonry,  and 
earth,  and  the  slope  of  the  beds  varied  from  .001  to  .10. 

The  results,  for  the  rectangular  and  trapezoidal  sections, 
sensibly  agreed  with  the  calculations  obtained  from  the  formula 


"but  with  circular  and  egg-shaped  sections  the  calculated  are 
about  10  per  cent  less  than  the  actual  results. 
In  practice  it  is  most  convenient  to  take 


v  =  ~    I/mi  =  c 


where  b  =  —  =  a  -I  ---  . 
c2  rm 

(Y  and  ft  are  not  constant,  but  have  values  depending  upon 
the  character  of  the  channel  faces  and  bed.  Bazin  gives  the 
following  table: 


Character  of  the  Wetted  Surface. 

Value  of  a,  the  Unit  being 

Value  of  |8. 

A  Foot. 

A  Metre. 

Smooth  cement,  planed  wood,  etc  
Cut  masonry    bricks    planks             .    .. 

.000046 
.000058 
.000073 
.000085 
.OOOI2 

.00015 
.00019 
.00024 
.00028 
.00040 

.0000045 
.0000133 
.OOOO6 
.00035 
.0007 

Rubble  masonry     

Earth  .            

Boulders  (  Kutter)  

Tables  at  the  end  of  the  chapter  give  the  values  of  the 
coefficients  b  and  cy  a  metre  being  the  unit. 

Reviewing  the  results  of  more  than  700  experiments  carried 
out  in  France,  Europe,  the  United  States,  and  British  India, 


250  GANGUILLET  AND  KUTTEKS    FORMULA. 

etc.,  upon  canals  and  rectangular,  trapezoidal,  semicircular, 
and  circular  aqueducts,  of  different  dimensions,  Bazin,  in  1897, 
(Ann.  des  Fonts  et  Chaussees^)  deduced  the  formula 

I57-6 


87 
for  v  =  —          -  Vmi,  if  a  metre  is  the  unit). 

X  +  -^= 

1/m 

This  equation,  again,  may  be  most  conveniently  written  in 
the  form 

v  =  c  ^mi, 

and  Tables  at  the  end  of  the  chapter  give  the  values  of  c  for  the 
six  different  classes  into  which  Bazin  has  divided  all  channels, 
the  corresponding  values  of  the  coefficient  y  being  given  by 
the  following  table: 


Class. 

Character  of  the  Wetted  Surface. 

y,  the  Unit  being 

A  Foot. 

A  Metre, 

1. 
II. 
III. 
IV. 
V. 
VI. 

.109 
.290 
•833 
1.540 

2-355 

3.170 

.06 
.16 
.46 

•85 
I  30 

1.7  i 

Earthen  channels  or  rivers,  presenting  exceptional 
resistance  ;  the  beds  covered  with  boulders  and 

8.  Ganguillet  and  Kutter's  Formula. — Bazin's  is  the  only 
formula  used  in  France,  but  in  England,  Germany,  and  the 
United  States  engineers  prefer  the  formula  of  Ganguillet  and 
Kutter,  viz., 

v  =  c  ^mi, 
the  value  of  c  being  given  in  a  Table  at  the  end  of  the  chapter. 


GANGUILLET  AND  KUTTERS  FORMULA. 
Also  the  coefficient 


251 


c  = 


a,  1,  and  p  being  certain  constants  and  n  a  coefficient  depend- 
ing only  on  the  roughness  of  the  channel  sides  and  bed. 

If  the  unit  is  &foot,  a=4i.6;   1—  1.8112;  p=  .00281. 

If  the  unit  is  a  metre,  a=23;  1=  i;  p  =  .00155. 

The  unit  being  a  foot,  n  varies  from  .008  to  .05  and  the 
following  table  gives  the  values  of  n  which  will  be  found  of 
most  use  in  practice: 


Character  of  Sides. 

n. 

Authority. 

ooo 

-] 

Smooth  cement  

.01 

A  mixture  of  2  of  cement  to  I  of  sand 

Rough  planks  .. 

Ashlar  or  brickwork               ... 

OTO 

Canvas  on  frames          

.<-»!  j 

OI  £ 

Rubble  masorrv  

OI7 

Rivers  and  channels  in  very  firm  gravel.  ...           -  .  . 

.02 

Ganguillet 

"         "  perfect   order,    free    from  'de- 
tritus (stones,  weeds,  etc.). 
"  moderately  good    order,    not 
quite  free  from  detritus  or 
weeds     .  .       

.025 

o^ 

and  Kutter 

"        "  bad   order,    with    weeds   and 
detritus    

QOC 

OC 

Canals  in  earth  above  the  average  order 

Q22C 

-x 

"        "       "      in  fair  order  

,02s; 

"        "       "      below  the  average  order  

.027^ 

J-  Jackson 

"        "       "      in  rather  bad  order,  overgrown  with 
weeds  and  covered  with  detritus.  . 

•03 

The  difficulty  of  properly  selecting  the  value  of  n  is  due  to 
the  fact  that  there  is  no  absolute  measure  of  the  roughness  of 
channel  beds. 

In  obtaining  the  above  results  Ganguillet  and  Kutter  made 
a  careful  study  of: 


25 2  FORMULA  OF  MANNING,  TUTTON,  ETC. 

(a)  The  Experiments  of  Darcy  and  Bazin. — These  show 
that  c  depends  both  upon  the  roughness  and  the  sectional 
dimensions.  The  values  of  OL  and  ft  in  Bazin 's  formula  vary 
with  the  character  of  the  channel  sides  and  bed ;  but  while  in 
small  channels  the  influence  upon  the  flow  of  differences  of 
roughness  must  be  very  great,  it  is  certain  that  this  influence 
diminishes  as  the  sectional  area  increases,  and  that  it  will  be 
nil  when  the  area  is  infinitely  great. 

(<£)  The  Measurements  of  Humphreys  and  Abbot  on  the 
Mississippi,  a  stream  of  very  large  sectional  area  with  a  bed  of 
very  small  slope. 

(c)  Their  own    Gaugings   in     the    regulated   channels    of 
certain    Swiss   torrents   with    exceptionally   steep    slopes    and 
running  through  extremely  rough  channels. 

(d)  The  Effect  of  the  Slope. — The  coefficient  c  diminishes 
as  the  slope,  /,  increases.      The  value  of  c  does  not  vary  much 
with  the  slope  of  the  bed  in  small  rivers,  but  in  large  rivers 
with  small  slopes  the  variation  is  considerable. 

9.  Formulae  of  Manning,  Tutton,  Humphreys  and  Abbot, 
and  Gauckler. — In  1 890,  Manning  proposed  the  formula 

v  =  cjvfi.fi  =  -     —#****,   if  the  unit  is  a  foot, 
or 

v  =  cjrifi*  =  —m^i^y  if  the  unit  is  a  metre. 

In  this  formula,  which  gives  good  results,  the  coefficient  n 
has  the  same  value  as  the  n  in  Kutter's  formula. 
Bazin's  and  Manning's  formulae  are  identical  if 

c  \'mi  =  ~,-r  Vmi  =  ^fftb'}, 
i.e.,  if 

,=  -L  =  ^. 


EXAMPLES.  253; 

By  an  independent  method,  Tutton,  in  1893,  deduced  the 
corresponding  formula, 


n  being  again  the  same  as  in  Kutter's  formula. 

As  a  result  of  observations  on  the  Mississippi  in  1865, 
Humphreys  and  Abbot  deduced  the  rather  complicated 
formula 

v  —  {3  -873  (*»'**)*  —  -0388[  3,  the  unit  being  a  foot, 


v  =  \(6gm'i^  —  .O2I4J2,  the  unit  being  a  metre. 


or 


In  this  expression,  which  is  of  especial  value  for  large 
watercourses,  m'  is  the  ratio  of  the  sectional  area  to  the  total 
perimeter. 

Gauckler's  formulae  for  canals, 

v  =  cm*i,  if  the  slope  is  <  7  per  I  oooo, 
and 

v  =  c'mh't,    "          "        >  7  per  10000, 

and  Hagen's  formula, 

v  —  2. 4  3  *««**, 

the  unit  in  each  case  being  a  metre,  have  not  been   used  in 
practice. 

Ex.  i.  A  channel  with  a  fall  of  i  in  10,000  has  brickwork  faces,  is  of 
rectangular  section,  20  ft.  wide,  and  is  to  convey  200  cu.  ft.  of  water  per 
second.  What  must  be  the  depth  of  the  water  ? 

Let  x  be  the  required  depth.     Then 

A  =  20.*;     P  =  20  +  2,v, 

IO.T 
and  m  =  - 

10  +  x 


2. 54  EXAMPLES. 

Also, 


200 
20JT 


10  ./      10*  I  C      ./      \VX 

=  —  —  if  =  c  {/ .  —    —  =  — !/ . 

X  10    +   X      lOOOO          100  V      10    +   X      , 


This  equation  can  be  best  solved  by  trial. 
Let  x  .  =  5  ft.     Then 

'»  =^r  =  3-33  ft-, 


;and  the  Tables  give  127.2,  as  the  corresponding  value  of  c. 
Therefore 


v  =  loo"  i/3'333  =  2'3223  ft<  per  sec" 
and  Q  =  iooz/  =  232.23  cu.  ft.  per  sec., 

which  is  too  great. 

Let  x  =  4  ft.     Then 

m  =  —  =  2  8  c 
14 

and  the  Tables  give  126.4  as  tlie  corresponding  value  of  c. 
Therefore 


126.4  A/ 40 

It     =    —        —  I/          -       =     2.136;     ft. 
I  /-w-v        '  T    1  •/       ^ 


per  sec. 
loo    '     14 

and  Q  =  Sot/  =  170.92  cu.  ft.  per  sec., 

which  is  too  small. 

Thus  x  must  lie  between  4  and  5  ft.     Try  x  =  4.5  ft.     Then 


. 

14.5 

and  the  corresponding  value  of  r  is  126.8. 
Therefore 

126.8  .790 

v  =  —  —\   '•  —  =  2.2338  ft.  per  sec., 
loo   r    29 

and  <2  =  90^7  =  201.042  cu.  ft., 

which  is  very  nearly  correct.     By  further  trials  the  depth  can  be  ob- 
tained within  a  fraction  of  an  inch. 


EXAMPLES.  255 

Ex.  2.  A  canal  in  earth  with  sides  sloping  at  40°  is  to  convey  100 
cu,  ft.  of  water  per  sec.,  at  a  velocity  of  i  ft.  per  second.  What  is  the 
fall  of  the  canal,  and  what  are  its  most  suitable  dimensions? 

A  =  ioo.     Then  (see  Table,  p.  233), 

bottom  width  =  .525  j/ioo  —  5.25  ft., 
depth  of  water  =  .722  4/100  =  7.22  ft., 
mean  depth,  m  —  .361  |7ioo  =  3.61  ft. 

By  the  Tables  the  corresponding  value  of  c  is  93.3.     Therefore 


=  93-3  4/3- 6| 


and 


3H24 


Ex.  3.  A  length  of  the  La  Roche  cut  is  in  compact  rock.  Its  bottom 
width  is  0.70  m.,  the  depth  of  the  water  is  0.50  m.,  one  bank  is  vertical 
and  the  other  slopes  at  26°  34'  to  the  vertical.  If  the  fall  is  i  in  500, 
find  the  mean  velocity  and  quantity  of  flow. 

The  width  of  section  at  the  surface  =  .70  +  .50  tan  26°  34'  =  om.95. 

A  —  -(.95  +  .70).  50  =  0.4125  sq.  m. 

P  —  .50  -f  .70  +  .50  sec.  26°  34'  =  1.759  m- 

Therefore 

.4125 


1-759 
and  the  corresponding  value  of  b  in  the  Tables  is  .0423.     Hence 


.0423  x  TV  =  y  .2345  x  —  =  .02165, 

and  v  =  .512  m.  per  sec. 

Therefore,  also, 

Q  =  .512  x  .4125  =  .2112  c.m.  per  sec. 
Again,  using  Bazin's  formula  for  the  filament  of  max.  vel., 

z'max.  =  v  +  14  |/;///=  .512  +  14  x  .02165  =  om.8i5  per  sec., 
and  -vb  —  bottom  velocity  =  — (2/max.)  =  om.489   per  sec. 


256  EXAMPLES. 

Ex.  4.  In  another  length  of  La  Roche  cut,  in  earth,  the  banks  slope 
at  45°,  the  bottom  width  is  0.3  m.,  and  the  depth  of  the  water  is  0.5  m^ 
Find  the  coefficients  b  and  c,  the  discharge  being  .21  12  cu.  ft.  per  second, 
and  the  fall  i  in  500. 

A  =  -(1.3  +  -5).  5  =  o-4  sq.  m. 
P  =  .3  +  2  \'7$  =  rn.  7  14- 

Therefore  m  =  ~     -  =  .23337. 

1.714 

->  j  j  2 

Also,  v  =  -     —  =  om.528  per  sec. 

•  4 

Hence 


=  c  x  .02162  =  —  —  x  .02162 

\'b 

and  c  —  24.4,         b  =  .00168, 

which  closely  agree  with  the  results  given  by  the  Tables. 

Ex.  5.  Find  the  quantity  of  water  conveyed  by  a  channel  of  trape- 
zoidal section  lined  with  brickwork  and  having  a  fall  of  6  in  looo.  The 
water-surface  width  is  7.185  ft.,  the  bottom  width  is  6.56  ft.,  and  the 
depth  of  the  water  is  4.92  feet. 

A  =  —(7.185  +  6.56)   x  4.92  =  33.813  sq.  ft., 
P  =  6.  56  +  2  x  4.954  =  16.468  ft. 

Therefore  m  =  fgf  =  2.053. 

Hence 


Q  =  Av  =  c  x  33.8I3J/  2.053  x  ----  =  c  x  3.7528. 


For  m  =  2.053 

Baziri's  Tables  give  c  =  124.6,  and  then  Q  =  467.6  cu.  ft.  per  sec. 
Manning's        "         "     c  =  128.6,  and  then   <2  —  482.6       "  " 

Kutter's        "         "     c  —  130.4,  and  then  Q  —  489.36     "  " 

and  the  differences  in  the  three  cases  are  not  considerable. 


VELOCITY   VARIATION  IN   TRANS  YERSE  SECTION.          257 

10.  Variation  of  Velocity  in  the  Transverse  Section  of  a 
Watercourse. — The  discharge  (Q)  across  any  transverse  sec- 
tion of  a  watercourse  is  the  product  of  the  area  (A)  of  the 
section  and  the  mean  velocity  (v)  of  flow.  Thus 

Q=Av. 

The  value  of  v  for  channels  of  small  section  can  easily  be 
found  by  discharging  into  a  suitable  reservoir  for  a  definite 
interval  of  time,  when  Q  can  be  estimated ;  and  since  A  is 
known,  v  can  be  at  once  calculated.  This  method  is  imprac- 
ticable with  watercourses  of  large  dimensions.  The  profile  of 
the  section  must  then  be  carefully  plotted,  when  its  area  can 
be  obtained  with  a  planimeter  or  by  the  method  of  mean 
heights.  The  velocity  of  flow  varies  from  point  to  point 
throughout  the  section  in  a  most  irregular  manner,  and  its  value 
has  not  been  fixed  by  any  single  law.  By  using  a  meter  or 
gauge  the  velocity  may  be  measured  at  a  large  number 
of  points,  and  in  this  manner  the  mean  velocity  (v}  and 
the  maximum  velocity  (s>max.)  can  be  very  approximately 
determined.  The  velocity,  however,  varies  so  much  and 
depends  so  largely  upon  the  conditions  under  which  the  flow 
takes  place,  that  it  seems  hopeless  to  expect  that  the  compli- 
cated law  of  velocity  distribution  can  be  expressed  in  a  general 
formula.  The  numerous  experiments  of  Bazin  on  the  Bour- 
gogne  canal  and  on  the  Seine  and  Saone,  of  Cunningham  on 
the  Ganges  canal,  and  of  Humphreys  and  Abbot  on  the 
Mississippi,  all  go  to  prove  this  and  at  the  same  time  throw 
much  light  upon  the  whole  subject.  It  has  been  shown  that  the 

ratio  —   -   diminishes  as  the  resistance  of  the  sides  and  bed, 

which  is  measured  by  the  expression  — ,  increases.     The  ratio, 

for  example,   is  about   .85   in  a  channel  with  a  very  smooth 
surface  and  falls  to  about  .  50  when  the  channel  is  cut  through 


258  VELOCITY   VARIATION  IN   TRANSVERSE  SECTION. 

earth.  As  the  surface  resistance  diminishes  the  value  of  — ^ 
tends  to  become  very  small  and  ultimately  zero,  while  the 
ratio  -  —  tends  to  become  unity.  Bazin  therefore  expressed 
the  relation  between  v^..  and  v  in  the  form 


-+'(3) 


in  which  the  function  /M— H  vanishes  with  — ,. 

\  v2 1  v* 

A  special  case  is  Bazin 's  empirical  formula, 


(3) 
(4) 


the  values  of  b  and  <:  being  given  by  the  Tables,  and  K  being 
a  coefficient  depending  upon  the  form  of  the  section  and  the 
conditions  of  flow.  For  example,  if  ^'max.  is  the  maximum 
surface  velocity  for  a  given  section, 


V 

for  a  watercourse  of  great  width  as  compared  with  the  depth 
and 


for  a  channel  of  restricted  dimensions,  as  in  ordinary  practice. 

Again,  if  ^max.  is  the  maximum  velocity  for  the  whole  sec- 

tion of  such  a  channel,  and  if  vm  is  the  mean  velocity  along 


VELOCITY  VARIATION  IN    TRANSVERSE  SECTION.          259 

the    vertical     in    which     the     maximum     velocity    lies,    then, 
approximately, 


(7) 


in  which  h  is  the  depth  of  the  water  on  the  vertical  in  ques- 
tion. (If  a  metre  is  the  unit,  the  values  of  K  in  the  three  last 
formulae  are  20,  14,  and  6,  respectively.) 

For  channels  of  mean  dimensions  Prony  has  suggested  the 
formula 

*'  7-7%  +  ^max.  , 

'~' 


(If  the  unit  is  a  metre,  substitute  2.37  for  7.78,  and   3.15  for 

10.34-) 

In  the  same  case  Dubuat  gives 


(9) 


in  which  vb  is  the  velocity  at  the  bottom  of  the  channel. 

For  values   of  z>'max.  up  to  about  11  or  12  ft.  (3.5  m.)  per 

v 

second  the  calculated  values  of  the  ratio  —  vary  but  little  from 

v 

the  average  value  .8,  a  result  which  has  been  verified  in  certain 
special  experiments.  It  is  therefore  considered  sufficient  to 
take 


and  then,  by  eq.  (9), 


When  the  water  is  of  great  depth  the  ratio  —  --  falls  to 

^  max. 

75,  and  to  .60  if  the  bottom  is  covered  with  reeds. 


260  VELOCITY   VARIATION  IN    TRANSVERSE  SECTION. 

Sonnet  has  theoretically  deduced  for  watercourses  of  great 
width  the  relation 


so  that  if  v  =  l^'max.*  then  v6  =  %v'msatm 

For  a  long  time  it  was  supposed  that  the  maximum  velocity 
(^max)  was  m  the  free  surface,  and  its  value  was  determined  by 
observing  the  time  in  which  floats  passed  between  two  trans- 
verse sections  at  a  specified  distance  apart.  Experiments  have 
now  demonstrated  that  this  maximum  velocity  is  at  some  point 
below,  although  in  general  near  the  free  surface,  and  the  floats 
will  not  give  the  proper  value  of  the  maximum  velocity  unless 
they  are  suitably  submerged.  It  has  also  been  found  that  the 
depth  of  this  point  of  maximum  velocity  increases  as  the  ratio 
of  the  width  to  the  depth  of  the  waterway  diminishes,  and  may 
be  as  great  as  one  third  of  the  depth  of  the  water. 

On  any  horizontal  line  at  right  angles  to  the  axis  of  the 
channel  the  velocity  diminishes  with  the  depth  of  the  water, 
is  greatest  towards  the  centre,  and  diminishes  at  an  increasing 
rate  on  approaching  the  sides. 

The  experiments  of  Darcy  and  Bazin  have  shown  that  the 
air-resistance  is  not  the  most  important  factor  in  causing  the 
variation  in  the  velocity  throughout  the  section.  With  a 
gauge  they  determined  the  velocities  at  a  number  of  points  in 
the  cross-section,  and  plotted  the  corresponding  equal-velocity 
curves  : 

(a)  For  a  closed  wooden  pipe,  of  rectangular  section, 
running  full  (Fig.  141); 

(/£)  For  an  open  wooden  channel  running  half  full  and 
formed  by  removing  the  upper  side  of  the  pipe  in  (a)  (Fig. 
142). 

The  curves  for  the  pipe  are  approximately  rectangular  and 
parallel  to  the  sides  of  the  pipe.  The  discharge  in  the  open 
channel  is  slightly  greater  than  one  half  of  the  pipe's  discharge, 


VELOCITY   VARIATION  IN   TRANSVERSE  SECTION. 


261 


but  there  is  no  similarity  between  the  equal-velocity  curves  in 
the  two  cases.  In  the  open  channel  they  become  more 
-elliptical,  tend  to  close  at  the  centre,  and  cut  the  free  surface 
obliquely,  the  angle  of  incidence  becoming  more  and  more 
acute  towards  the  centre.  The  curves  are  also  at  a  greater 
distance  from  the  centre  than  the  corresponding  curves  in  the 
pipe.  This  very  marked  modification  in  the  form  of  the 


FIG.  141. 


FIG.  142. 


velocity  curves  is  due  especially,  in  Bazin's  opinion,  to  the 
absence  of  the  upper  boundary  and  to  the  consequent  practical 
impossibility  of  an  absolutely  constant  cross-section.  Eddies 
and  other  irregular  movements  are  .produced  in  the  surface  and 
give  rise  to  corresponding  losses  of  energy  and  velocity. 
Actual  experiment,  too,  has  shown  that,  even  with  a  strong 
wind  blowing  down-stream,  tending,  as  might  be  supposed,  to 
cause  an  excessive  surface  velocity,  the  maximum  velocity  is 
still  at  some  point  below  the  free  surface. 

For  any  given  vertical  in  the  section  it  appears  to  be 
approximately  true  that  the  velocity  at  about  three  fifths  of  the 
total  depth  is  sensibly  the  mean  velocity  for  the  whole  depth, 
and  that  the  difference  between  the  maximum  and  bottom 
velocities,  viz.,  7'max.  —  v&j  increases  with  the  roughness  and  lies 
between  i^max.  and  <^max>. 

In  a  semicircular  channel  of  radius  r  the  equal-velocity 
curves  are  circular,  Fig.  143,  and  concentric  with  the  bed,  the 


262          VELOCITY  VARIATION  IN  TRANS  I/ERSE  SECTION. 
velocity  v  at  the  distance  y  from  the  centre  being  given  by 

38  Vn  ' 


being  the  velocity  at  the  centre. 


FIG.  143. 

Generally  speaking,  the  equal-velocity  curves  are  approxi- 
mately of  the  same  form  as  the  profile  of  the  section  (Figs. 
143  to  146),  and  this  is  especially  the  case  near  the  sides  and 


FIG.  144. 


FIG.  146. 


bed.      The  curves  at  the  bottom  do  not  always  reach  the  sur- 
face, but  sometimes  cut  the  sides. 

Again,  experiments  indicate  that  the  law  of  velocity  dis- 
tribution along  any  vertical  in  the  section  may  be  represented 
by  a  parabola  of  the  2d  degree,  with  its  axis  horizontal  and  at 
the.  same  depth  as  the  point  of  maximum  velocity.  Defontaine 
in  an  experiment  on  an  arm  of  the  Rhine  deduced  for  the 
vertical  at  the  centre  of  the  current  the  analogous  law 

u  —  4.8222  —  .066^,       ....      (13) 

u  being  the  velocity  at  the  depth  y. 

(If  the  unit  is  a  metre,  u  =  1.266  —  .252^.) 


VELOCITY  VARIATION  IN   TRANSVERSE  SECTION.          263 

The  following  theoretical  investigation  of  the  velocity  curve 
&  based  on  the  assumptions  that : 

(a)  The  watercourse  is  of  very  great  width  as  compared 
with  the  depth ; 

(b)  The  watercourse  is  of  sensibly  uniform  depth ; 

(c)  The  fluid  particles  flow  across  a  transverse  section  in 
sensibly  parallel  lines ; 

(d)  A  permanent  regime  has  been  established  so  that  the 
pressure  is  distributed  over  the  section  in  sensibly  parallel  lines ; 

(e)  The  resistance  to  the  relative  flow  of  consecutive  fluid 
filaments  is  of  the  nature  of  a  viscous  resistance. 

Let  Fig.  147  represent  a  portion  of  a  vertical  longitudinal 


1 

ct|— 

I 

j 

>,!  

dy 

. 
j 

c 

*~- 
1 

-i 

>  > 

B 

J 

5 

FIG.  147- 

section  of  the  stream  intersected  by  two  transverse  sections 
AB,  CD,  /  being  the  distance  between  them. 

Consider  a  thin  layer  abed  of  thickness  dy  and  width  b, 
bounded  by  the  sections  AB,  CD,  and  by  the  planes  ad,  be, 
at  depths  y  and  y  +  dy,  respectively,  below  the  free  surface. 

The  forces  acting  upon  the  layer  in  the  direction  of  motion 
are: 

(1)  The   pressures   on   the   ends   ab,   cd,   which  evidently 
neutralize  each  other. 

(2)  The    component    of   the    weight  =  wbl.dy.sm    i  = 
wbli .  dy;  i  being  the  slope  of  the  bed. 

(3)  The  viscous  resistances  on  the  lateral  faces  of  the  layer 
under    consideration.      These    are    nil,    since   in   a   stream    of 
indefinite  width  there  will  be  no  relative  sliding  between  abed 
and  the  vertical  faces  on  each  side. 


264          VELOCITY   VARIATION  IN   TRANSVERSE  SECTION. 

(4)  The  viscous  resistances  along  the  planes  ad  and  be. 

The  frictional  resistance  to  distortion,  i.e.,  to  shearing, 
along  such  planes,  is  found  to  be  proportional  to  the  shear  per 
unit  of  time,  and  is  measured  by  the  shear  per  unit  of  area  at 
the  actual  rate  of  shearing.  The  coefficient  of  viscosity,  or 

i      ,,         •  i  shear  per  unit  of  area 

simply  the  viscosity,  is  the   quotient : , 

shear  per  unit  of  time 

and  defines  that  quality  of  the  fluid  in  virtue  of  which  it  resists 
a  change  of  shape. 

Adopting  Navier's  hypothesis, 

the  viscous  resistance  along-  ad  =  —  kbl — , 

dy 

k  being  the  coefficient  of  viscosity,  and  u  the  velocity  at  the 
depth  y.      The  sign  is  negative  as,  since  u  increases  with  y, 

-j-  is  positive,  and,  at  the  same  time,  the  action  of  the  layers 
above  ad  is  of  the  character  of  a  retardation. 

The  viscous  resistance  along  be  =  kbl—  4-  kbl    d(—\ 

dy  \dy) 


dy*         dy^ 
Then,  as  the  motion  is  uniform, 

wbli .  dy  -  kbl-  +  kbl^  +  kbl^dy  =  o. 
dy  dy   '         dy*  ' 

Hence 

wi 


Integrating  twice, 

wt 


(14) 


a  and  vs  being  constants  of  integration 


VELOCITY  VARIATION  IN   TRANSVERSE  SECTION.          265 

It  is  evident  that  vs  is  the  surface  velocity,  i.e.,  the  value  of 
#*when  y  =  o. 

The  equation  may  be  written  in  the  form 


2WI 


wi 

2~k 


Thus   the   velocity  curve   is    a    parabola 

ka 

having  a  horizontal  axis  at  a  depth  F  =  — . 

wi 

below    the    free   surface.      This   is    also    the 
depth   of  the  filament  of  maximum  velocity 

idu         \ 

I  -j-  —  °     and 
\dy         / 

h^  W 

*v.  +  -,y*.    (16) 


FIG.  148. 


Hence,  by  equations  (14)  and  (16), 


w 


Let  vm  be  the   ' '  mean  ' '  velocity  for  the  whole  depth  h. 
Let  z/i  be  the  mid-depth  velocity.      Then 


.     .     .     (18) 


and 


i  lh 


266          VELOCITY  VARIATION  IN   TRANSVERSE  SECTION. 

Hence 

wit? 


a  result  upon  which   Humphreys  and   Abbot    have    based   a 
rapid  method  of  gauging  rivers. 

Let  v6  be  the   bottom  velocity,  i.e.,  the  value  of  v  when 
y  =  h.     Then,  by  equation  (17), 


w 


and  therefore, 


wi 

^max.  -  Vb  =  -(k  -   Vy   =  N,     SUPPOSe.     .        .        (2  l) 


According  to  Bazin,  ^max.  —  v6  is  sensibly  constant  and  is 
approximately  equal  to  36.  3  Vki  (=.  20  1//«  if  a  metre  is  the 
unit).  Thus  the  general  equation  (15)  of  the  velocity  curve 
becomes 


«  =  *W.-36.3y>**\j^-y-j.      •     •     •     (22) 

This,  known  as  Bazin 's  formula,  agrees  well  with  the 
experiments  on  artificial  channels  and  on  the  Saone,  Seine, 
Garonne,  and  Rhine.  It  was  found,  in  general, 

that  -^^  =  1 .  1 7  in  the  Rhine  at  Basle  and  ranged  from  I .  I 
to  1.13  in  the  other  channels; 


8   ran^ed  from  J  3  to  20; 


-  F) 


VELOCITY  VARIATION  IN   TRANSVERSE  SECTION.          267 

I 

- 
j 


F        I 
.m   -r  =  -  in  some  artificial  channels  and  in  others  ranged. 


from  o  to  .2. 

These  last  results  are  not  in  accord  with  the  Mississippi 
measurements. 

In  the  case  of  a  rectangular  channel  of  such  width  that  the 
influence  of  the  sides  on  the  flow  may  be  disregarded,  the 
mean  radius,  m,  may  be  substituted  for  h  and  the  mean 
velocity,  vm,  is  sensibly  the  same  as  the  mean  velocity,  v,  for 
the  whole  section.  Hence  equation  (22)  may  be  written 


v 
or 


,  (23) 

— 


the  value  of  b  being  given  by  the  Tables. 

Filament  of  Maximum    Velocity  in  the  Surface. — In  this 
case  Y  —  o,  and  equation  (21)  becomes 

wi 

V  max.  ~  V6  =   ^ (24) 

^ 'max.  being  the  value  of  z/max.  when  the  maximum  velocity  is 
in  the  surface. 

Equation  (22)  also  becomes 


-.        •        •        •        (25) 

Again,  by  equations  (18)  and  (24), 


w 


268          VELOCITY  VARIATION  IN  TRANSVERSE  SECTION. 
or 

^'max.H-  V*  ,    „ 

V~=-       —  --  »  .......        (26) 


result  already  referred  to. 

Boileau  s   Formula.  —  Boileau    assumes    that   the    velocity 
•o  _______  v._  _____  Mt  _N_  curve  is  given  by  the  equation 

Cy    .     .     (27) 


„. 


M 

above  the  point  of  maximum  velocity,  and 
below  this  point  by   the  equation 

u=D-B?  .....     (28) 


FIG.  149.  When  y  =  o,      u  =  vt  =  A  . 

When  y=Y, 

D  _  BY*  =  z/max.  =  A  -  BY*+CY=  vs  -  BY*  +  CY. 
Therefore 


Y 

Boileau  's  experiments  led  him  to  infer  that  the  difference 
D  —  z>max.  (—  £Y2)   is    sensibly  v  constant.       Designating    this 

difference  by  d,  so  that  D  =  ^max.  +  d  and  B  =  pr2,  Boileau  's 
equations  become 

^max.  —  ^  +  d  y 


.        .        (29) 

representing  the  curve  MM2  ,  and 

«  =  ^.+^~XF)  .......  (30) 

representing  the  curve  J^fr 


TABLES  OF  EROSION  4ND   VISCOSITY. 


269 


ii.  Tables  of  Erosion  and  Viscosity. 

-..TABLE    INDICATING   THE   VELOCITIES    ABOVE   WHICH 
EROSION   COMMENCES. 


Nature  of  the  Channel  Bed. 

z 

i 

V 

m 

i) 

b 

Met. 

Feet. 

Met. 

Feet. 

Met. 

Feet. 

if\ 

•J5 

•49 

r»R 

•  3° 

,fi 

.  20 

•3° 

f\n 

•  9» 

•23 

Af\ 

•75 

•52 

I.Q7 

.40 

r>f- 

I.51 

•31 

.90 

3-  T5 

.70 

2.30 

_    r>9 

Soft  schist  

1.52 

5-OO 

7    28 

1.23 

i  86 

4-°3 

•94 

3.08 

Stratified  rocks            .  • 

•49 

T      So 

4.90 

Hard  rocks  

•  75 
427 

2.27 

7-45 

•UV 

•  *4 

1U.  J 

*  TABLE   OF    VISCOSITIES    (Everett's  System   of  Units). 


WATER. 

MERCURY. 

Temp. 
(Cent.) 

Viscosity. 

Temp. 

(Cent.) 

Viscosity. 

Temp. 
(Cent.) 

Viscosity. 

Temp. 

(Cent.) 

Viscosity. 

O° 

.Ol8l 

35° 

.0073 

0° 

.0169 

3I50 

.00918 

5 

.0154 

40 

.0067 

10 

.0162 

340 

.00897 

10 

•0133 

45 

.0061 

18 

.0156 

15 

.OIT6 

50 

.0056 

99 

.0123 

20 

.OI02 

60 

.0047 

154 

.0109 

25 

.009! 

80 

.0036 

197 

.OI02 

30 

.008l 

90 

.0032 

249 

.  00964 

The  viscosity  is  given  by 


' ^cording  to  Meyer, 


and  by 


5212 


'^        " .001  31,  according  to  Slotte; 


26 


/  being  the  temperature  centigrade. 

12.  River-bends. — The  following  explanation  is  due  to 
Professor  James  Thomson  (Inst.  Mechl.  Engs.,  1879;  Proc. 
Royal  Soc.  1877).  In  rivers  flowing  in  alluvial  plains,  the 


*  N.    B.   The  viscosities  are  in  C.   G.  S.   units, 
units  and  centigrade  degrees  multiply  by  2.0481. 


To  reduce  to  F.  P.  S. 


270  RIYER-BENDS. 

curvature  of  the  windings  which  already  exist  tends  to  increase 
owing  to  the  scouring  away  of  material  from  the  outer  bank 
and  to  the  deposition  of  detritus  along  the  inner  bank.  The 
sinuosities  often  increase  until  a  loop  is  formed,  with  only  a 
narrow  isthmus  of  land  between  two  encroaching  banks  of  a 
river.  Finally  a  cut-off  occurs,  a  short  passage  for  the  water 
is  opened  through  the  isthmus,  and  the  loop  is  separated  from 
the  river-course,  taking  the  form  of  a  horseshoe  shaped  lagoon 
or  swamp.  The  ordinary  supposition,  that  the  water  always 
tends  to  move  forward  in  a  straight  line,  rushing  against  the 
outer  bank  and  wearing  it  away,  and  at  the  same  time  causing 
deposits  at  the  inner  bank,  is  correct,  but  it  is  very  far  from 
being  a  complete  explanation  of  what  takes  place. 

When  water  flows  round  a  circular  curve  under  the  action 
of  gravity  only,  it  takes  a  motion  like  that  in  a  free  vortex. 
Its  velocity  parallel  to  the  axis  of  the  stream  is  greater  at  the 
inner  than  at  the  outer  side  of  the  curve. 

Thus,  too,  the  water  in  a  river-bank  flows  more  quickly 
along  courses  adjacent  to  the  inner  bank  of  the  bend  than 


FIG.  150. 

along  courses  adjacent  to  the  outer.  The  water,  in  virtue  of 
centrifugal  force,  presses  outwards  so  that  the  water-surface  of 
a  transverse  section  (Fig.  150)  has  a  slope  rising  upwards  from 
the  inner  to  the  outer  bank.  Hence  the  free  level,  for  any 
particle  of  the  water  near  the  outer  bank,  is  higher  than  the 
free  level  for  any  particle  in  the  same  transverse  section  near 
the  inner  bank,  but  the  tendency  to  flow  from  the  higher  to 
the  lower  level  is  counteracted  by  centrifugal  action.  Now 
the  water  immediately  in  contact  with  the  bottom  and  sides  of 
the  course  is  retarded,  and  its  centrifugal  force  is  not  sufficient 


RIMR-BENDS. 


271 


to  balance  the  pressure  due  to  the  greater  depth  at  the  outside 
ofihe  bend.  This  water  therefore  tends  to  flow  from  the  outer 
bank  towards  the  inner  (Fig.  151),  carrying  with  it  detritus 


FIG.  151. 

which  'is  deposited  at  the  inner  bank.  Simultaneously  with 
the  flow  of  water  inwards,  the  mass  of  the  water  must  neces- 
sarily flow  outwards  to  take  its  place. 

13.  Flow  of  Water  in  Open  Channels  of  Varying  Cross- 
section  and  Slope. 

Assumptions. — (a)  That  the  motion  is  steady. 

Thus  the  mean  velocity  is  constant  for  any  given  cross- 
section,  but  varies  gradually  from  section  to  section. 

(b)  That  the  change  of  cross-section  is  also  gradual. 

(c)  That,  as  in  cases  of  uniform  motion,  the  work  absorbed 
by  the  frictional  resistance  of  the  channel  bed  and  sides  is  the 
only  internal  work  which  need  be  taken  into  consideration. 

Let  Fig.  152  represent  a  longitudinal  section  of  the  stream. 
The  fluid  molecules  which  are  found  in  any  plane  section  ab 
at  the  commencement  of  an  interval  will  be  found  in  a  curved 
surface  dc  at  the  end  of  the  interval,  on  account  of  the  different 
velocities  of  the  fluid  filaments. 


272 


CHANNELS   OF  VARYING   CROSS-SECTION. 


Suppose  that  the  mass  of  water  bounded  by  the  two  trans- 
verse sections  ab,  ef  comes  into  the  position  cdhg  in  a  unit  of 
time.  Then  the  change  of  kinetic  energy  in  this  mass  is  equal 
to  the  algebraic  sum  of  the  work  done  by  gravity,  of  the  work 


/   9 


done  by  pressure,  and  of  the  work  done  against  the  frictional 
resistance. 

Change  of.  Kinetic  Energy.  —  This  is  evidently  the  difference 
between  the  kinetic  energies  of  the  masses  efgh  and  abed, 
since,  as  the  motion  is  steady,  the  kinetic  energy  of  the  mass 
between  cd  and  ^remains  constant. 

Let  Al  be  the  area  of  the  cross-section  ab. 
"     #t   <(     "    mean  velocity  across  this  section. 
"     v    "     "    velocity   at   this   section  of  any  given   fluid 

filament  of  sectional  area  a. 
Let  v=ul±  V. 


Then 


A  fa  =  2(av)     and 


=  o. 


The  kinetic  energy  of  the  mass  abed 


=     -2\a(u*  ± 


3*1**  ± 


since 


=  o     and 


V— 


CHANNELS   OF  VARYING   CROSS-SECTION.  273 

Now    2?^  -|-  v   is   evidently    positive.      Hence    the   kinetic 
energy  of  the  mass  abed 


w 


>  —A.u* 

2g     l    l 
=  a  —  AM 


ct  being"  a  coefficient  of  correction  whose  value  depends  upon 
the  law  of  the  distribution  of  the  velocity  throughout  the 
section  ab.  It  is  positive  and  greater  than  unity.  Assume 
that  a  has  the  same  value  for  the  sections  ab  and  ef.  Then  if 
A.,,  ?/2  are  the  area  and  mean  velocity  at  the  transverse  section 
eft  the  kinetic  energy  of  the  mass  efgh 


. 

2g     *    * 

Hence    the   change   of  kinetic   energy  in   the    mass   under 
consideration 


wQ  u?  -  u* 

—   Oi  -  —  , 

g  2 

since  ^2  =  Q  =  A\ur 

Work  done  by  Gravity.  —  Consider  any  fluid  filament  inn, 
the  depth  of  m  below  the  surface  being  y^  ,  and  of  n,  y¥ 
Let  z  be  the  fall  in  the  surface-level  from  a  to  e. 
Then  the  fall  from  m  to  n 


and  the  work  done  by  gravity  on  the  elementary  volume  dQ 
in  a  unit  of  time 


274  CHANNELS  OF  VARYING   CROSS-SECTION. 

Work  done  by  Pressure. 

The  pressure  per  unit  of  area  at  m  =  wy^  -|~  /0  ; 
"      "     ,"      "      M   n  =  wy2  +  Pv 
pQ  being  the  atmospheric  pressure. 

Hence  the  work  due  to  these  pressures  per  unit  of  time 


Thus  the  total  work  done  by  gravity  and  by  pressure 


=  2(w  .  dQ  .  z)  =  wQs 

for  the  mass  under  consideration. 

Work   absorbed  by  Friction. — Consider   a   thin    lamina   of 
water  of  thickness  ds,  bounded  by  the  transverse   planes  xx, 
yy,  the  distance  of  xx  from  ab  being  s. 

Since  the  change  of  velocity  is  gradual,  the  mean  velocity 
from  xx  to  yy  may  be  assumed  to  be  constant. 

Let  u  be  this  mean  velocity. 
"   P  be  the  wetted  perimeter  at  the  section  xx. 
1  *   A  be  the  area  of  the  waterway  at  the  section  xx. 

Then  the  work  absorbed  by  friction  per  second  from  xx  to> 

yy 

—  P.ds.u.  F(u), 
and  the  total  work  absorbed  between  ab  and  ef 

=  <2 

/  being  the  distance  between  ab  and  ef.      Hence 


CHANNELS  &F  VARYING   CROSS-SECTION.  275 


and  therefore     ,  =  «*«'-*»'  .      (*  P 


w 


ds 


Take  =/          and          =  m.     Then 


„*_    M2  /•'/       2 

=a^  --  1_,        I     /  --  ^y         ^       m      ^ 

2&  Jo     Wig 


(I) 


If  the  two  planes  ab  and  cf  are  indefinitely  near  one  another 
(Fig.   153),  the  last  equation  evidently  gives 

dz  =  —u  .  du  -\ — ds,        .  -  (2) 

g  m  2g 

which  is  the  fundamental  differential  equation  of  steady  varied 
motion,  dz  being  the  fall  of  surface  level  in        ,_        a 
the  distance  ds. 


In  the  figure  aa   is  drawn  parallel  to  the         /^ 
bed  and  aa"  is  horizontal.      The  distance  ^x 
a  ' e  may,  without  sensible  error,  be  assumed 
equal  to  dz.  FIG.  153. 

Also,  a" a'  =  i  .  aa    =  i  .  ds,  very  nearly. 

Hence 

ids  =  a' a"  --=  a' e  -f  a"e  =  dh  +  dz,  .      .      .      (3) 

Substituting  the  value  of  dz  from  this   equation  in  equa- 
tion (2), 

/  .  ds  —  dh  =  —  u  .  du  +  —  —  .  ds.  ,  (A\ 

g  m  2g 

Also,  since  Au  =  Q,  a  constant, 

A  ,  du  +  u  .  dA  —  o, 
and  dA  =  x  .  dh.  very  nearly,  if  x  is  the  width  of  the  stream. 


276  CHANNELS   OF  YARYIHG   CROSS-SECTION. 

Therefore 

Adu  +  u*  •  dh  —  o, 
and  hence,  by  equation  (4), 


i.ds-  dh  =  -  *-—  .dh  +  —  u~ 

g  A  '    m  2g 


Therefore 


i  —  — 


dh  _  m  2g  .         m  2gi 

ds~~  ~^~x:  ~^'       ..-      (5) 

I  —  a  --  ~.  I  —  a  —  -. 
gA  gA 

Let  the  position  of  any  point  a  in  the  surface  be  defined  by 
its  perpendicular  distance  h  from  the  bed  and  by  the  distance 
j  of  the  transverse  section  at  a  from  an  origin  in  the  bed. 

Then  -r  is  the  tangent  of  the  angle  which  the  tangent  to  the 

surface  at  a  makes  with   the   bed.      It  is  positive  or  negative 
according  as  the  depth  increases  or  diminishes  in  the  direction 
of  flow,  thus  defining  two  states  of  steady  varied  motion. 
Between  these  there  is  an  intermediate  state  defined  by 

dh  f   w2 

—  -  =  o  =  i  —  -     —  , 
ds  m  2g 

f  "2 

and    i  =  ~        -    is    the    equation    for   steady  flow  with   uniform 
m2g 

motion. 

Let  U,  M,  H  be  the  corresponding  values  of  u,  m,  h  in  the 
case  of  uniform  motion.  Then 


~  —  =  b-M  ,       .....      (6) 
M  2g          M' 


.and  eq.  (5)  becomes 
dh 


ds                      zr  x                   u1  x 
I  —  a  —  I  —  a -: 

gA  g  A 


(7) 


CHANNELS   OF  VARYING   CROSS-SECTION. 


277 


If  the  section  of  the  channel  is  a  rectangle, 

xh 


'A  —  xh,   xhu  =  xHU,    m  = 


X  -\-.2k 
Substituting  these  values  in  eq.  (7), 


dh 


7,   and  M  = 


xH 


i  — 


Three  cases  will  be  considered  and,  in  each  case,  a  line 
PQ,  drawn  parallel  to  the  bed,  represents  the  surface  of 
-uniform  motion,  H  being  the  distance  between  PQ  and  the  bed. 

CASE  I.      an*  <  gh     and     H  <  //,  Fig.   154. 

-j-  is  positive,  and  therefore  h  increases  in  the  direction  of 

flow.      Thus  the  actual  surface  MN  of  the   stream  is  wholly 
above  the  line  PQ. 


FIG.  154. 

Proceeding  up-stream,  h  becomes  more  and  -more  nearly 
equal  to  //,  so  that  the  numerator  of  eq.  (8),  and  therefore  also 

dh 

j  ,  approximates  more  and  more  closely  to  zero. 

clS 

Again,  proceeding  down-stream,  h  increases  and  u 
diminishes,  so  that  both  the  numerator  and  denominator  in 
eq.  (8)  approximate  more  and  more  closely  to  the  value  unity* 

dh 
and  therefore    ,     becomes  more  and  more  nearly  equal  to  i* 

d-S 

the  slope  corresponding  to  uniform  motion. 


278  CHANNELS  OF  VARYING   CROSS-SECTION. 

Hence  up-stream  MN  is  asymptotic  to  PQ,  and  down- 
stream MN  is  asymptotic  to  a  horizontal  line.  This  form  of 
surface  is  produced  when  a  weir  is  built  across  a  channel  in 
which  the  water  had  previously  flowed  with  a  uniform  motion. 

CASE  II.      au*  <gk     and     H>  h,  Fig.   155. 

-r  is  now  negative,  and  the  depth  diminishes  in  the  direc- 
tion of  flow. 

Up-stream  //  increases  and  approaches  H  in  value,  so  that 
MN  is  asymptotic  to  PQ, 

Down-stream  h  diminishes,  n  increases,  and  therefore  the 

au2 
value  of  —.    is  more  and  more  nearly  equal  to  unity. 

Thus,    in   the   limit,   the   denominator  in   eq.    (8)   becomes 

zero,  and  therefore    ,  •  =  .00  .      Hence  theory  indicates  that  at 
ds 

a  certain  point  down-stream  the  surface  line  MN  takes  a  direc- 
tion which  is  at  right  angles  to  the  general  direction  of  flow, 
This  is  contrary  to  the  fundamental  hypothesis  that  the  fluid 
filaments  flow  in  sensibly  parallel  lines.  In  fact,  before  the 


FIG.  155. 

limit  could  be  reached  this  hypothesis  would  cease  to  be  even 
approximately  true,  and  the  general  equation  would  cease  to 
be  applicable.  This  form  of  water-surface  is  produced  when 
there  is  an  abrupt  depression  in  the  bed  of  the  stream. 

Fig.  156  shows  one  of  the  abrupt  falls  in  the  Ganges  canal 
as  at  first  constructed.  The  surface  of  the  water  flowing  freely 
over  the  crest  of  the  fall  took  a  form  similar  to  MN  below  the 
line  PQ  of  uniform  motion.  The  diminution  of  depth  in  the 


CHANNELS  OF  VARYING   CROSS-SECTION. 


279 


approach  to  the  fall  caused  an  increase  in  the  velocity  of  flow, 
with  the  result  that  for  several  miles  above  the  fall  a  serious 
erosion  of  the  bed  and  sides  took  place.  In  order  to  remedy 
this,  temporary  weirs  were  constructed  so  as  to  raise  the  level 


FIG.   156. 

of  the  water  until  the  surface  line  assumed  a  form  MN'  corre- 
sponding approximately  to  PQ.      In  some  cases  the  water  was 
raised  above  its  normal  height  and  a  backwater  produced. 
CASE  III.      au*  >  gh     and     H  <  h,  Fig.  157. 

—  is  negative  and  the  surface  line  of  the  stream  is  wholly 
-above  PQ. 


FIG.  157- 


dh 


If  //  gradually  increases,  u  diminishes  and  ^-    approximates 

to  —  i  in  value. 

If  h  gradually  diminishes,  it  approximates  to  H  in  value, 

•dh 

and  in  the  limit  -.    =  o. 
ds 


CHANNELS  OF  VARYING   CROSS-SECTION. 

Between  these  two  extremes  there  is  a  value  of  h  for  which 
the  denominator  of  eq.  (8)  becomes  nil,  viz., 

h  =  a— , 
g 

and  the  corresponding  value  of  ~j-  is  infinity. 

Thus  one  part  of  the  surface  line  is  asymptotic  to  PQ,  the 
line  of  uniform  motion,  another  part  is  asymptotic  to  a  hori- 
zontal line,  while  at  a  certain  point  at  which  the  depth  is 

h  =  a—  , 
g 

the  surface  of  the  stream  is  normal  to  the  bed. 

This  is  contrary  to  the  fundamental  hypothesis  that  the 
fluid  filaments  flow  in  sensibly  parallel  lines,  and  the  general 
equation  no  longer  represents  the  true  condition  of  flow. 

In  cases  such  as  this  there  has  been  an  abrupt  rise  of  the 
surface  of  the  stream,  and  what  is  called  a  "  standing  wave  " 
has  been  produced. 

In  a  stream  of  depth  H  flowing  with  a  uniform  velocity  [7, 


which 


IgH 

ich  is  >  A  /    — , 
V     a 


to  hv  which  is  > 


construct  a  weir  so  as  to  increase  the  depth 


d 

FIG.  158. 
Then  in  one  portion  of  the  stream  near  the  weir  the  depth 

aU* 
is  >  -  — ,   while  further  up  the  stream   the  depth  is  < 


STANDING    WAVE. 


281 


U* 
Thus  at  some  intermediate  point  the  depth  =  a — ,  the  corre- 

'-  £ 

sponding  value  of  -y-  being   oo  ,   and  at  this    point   a   standing 

wave  is  produced. 
Now 


=  Mi  =  Hi, 


U2 
and  since     H  <  ot  — , 


<  a — 


and  therefore 


2a 


which  condition  must  be  fulfilled  for  a  standing  wave. 
Bazin  gives  the  following  table  of  values  of/": 


Nature  of  Bed. 

Slope  |A  =  /) 

below  which  stand- 
ing wave  is  im- 
possible,    In 
Metres  per  Metre. 

Standing  Wave  Produced. 

Slope  in  Metres 
per  Metre  (or 
Feet  per  Foot). 

Least  Depth, 
in  Metres. 

£ 

Very  smooth  cemented  surface.  .  .  . 

.00147 
.00186 
.00235 
.00275 

(   .002 
-j   .003 
(   .004 
(  .003 
i  .004 
(  .006 
{.004 
.006 
.010 

(  .006 

-<    .010 

(  .015 

.08 
•03 
.02 
.12 
.06 
.03 
•36 
.16 
.08 
I.  O6 

•47 
.28 

Earth     ,  

A  standing  wave  rarely  occurs  in  channels  with  earthen 
beds,  as  their  slope  is  almost  always  less  than  the  limit,  .00275. 


282 


STANDING    WAVE. 


The  formation  of  a  standing  wave  was  first  observed,  by 
Bidone  in  a  small  masonry  canal  of  rectangular  section. 
The  width  of  the  canal  =  om.325  =  x\ 


slope  (=  j 


of  the  canal  =       .023; 


"     uniform  velocity  of  flow     =  im.69     =  U\ 
ft     depth  corresponding  to  U  =  om.o64  =  H. 
A  weir  built  across  the  canal  increased  the  depth  of  the 
water  near  the  weir  to  om.287  =  hr 

It  was  found  that  the  *  *  uniform  regime  ' '  was  maintained 
up  to  a  point  within  4m.  5  of  the  weir.  At  this  point  the  depth 
suddenly  increased  from  Om.o64  to  about  om.  170,  and  between 
the  point  and  the  weir  the  surface  of  the  stream  was  slightly 
convex  in  form  (Fig.  158). 

With  the  preceding  data   and  taking    a  =  I .  I , 


and  is  therefore  >   I  at  a  section  ab,  Fig.   159. 
At  the  section  cd, 


.69  =  0™. 377, 


and 


=  -O55 


therefore  < 


Thus  the  expression  I  --  —  is  negative  for  a  section  ab 

and  positive  for  a  section  cd,  so 
that  /  must  change  sign  between 

dh 

these  sections,  and      -  will  then 
ds 

become  infinite. 

Consider  a  portion  of  a  stream 
bounded  by  two  transverse  sections,  ab,  cdy  in  which  a  stand- 
ing wavs  occurs,  Fig.  159. 

Assume  that  the  fluid  filaments  flow  across  the  sections  in 
-sensibly  parallel  lines. 


FIG.  159. 


STANDING    WAVE.  283 

Let  the  velocities  and  area  at  section  ab  be  distinguished 
^y  the  suffix  i,  and  those  at  cd  by  the  suffix  2.      Then 

Change   of   momentum   in   di-  ) 

fa  I  —  impulse  in  same  direction, 

rection  of  flow  ) 


Hence 


and  therefore 

l-(2av*-2av*)  =  Alyl-A.i?2,     ...     (9) 

<*> 

yl  ,  j/2  being  the  depths  below  the  surface  of  the  centres  of 
gravity  of  the  sections  ab,  cd,  respectively. 
Now,  ^  =  u^  +  Vr      Therefore 


Also,  as  already  shown, 

arid,  neglecting  Vl  as  compared  with 

aAvu*  =  A^i* -\-  3« 
Thus 


and  hence 


a  +  2 
where  a'  =  ~       — ,  and  is  1.033  if  «  =  i  •  i . 


284  STANDING    WAVE. 

Similarly  it  may  be  shown  that 


Thus  equation  (9)  becomes 

'-Alu*)=Ajl-Ajr    .     .     .     (ID) 


Let  the  section  of  the  canal  be  a  rectangle  of  depth  Hl  at 
ab  and  H2  at  cd.      Then 

TT  TT 

tilHl  =  u2H2-     -f  =  y^    ~  =  yr 

Therefore,  by  equation  (10), 


which  reduces  to 


H2  =  Hl  satisfies  the  equation  and  corresponds  to  a  condition 
of  uniform  motion. 
Also, 

a'u?  *_H^  +  H, 

--  -~- 


In  Bidone's  canal,  «t  =  im.6g,  H^  =  om.o64.  Substituting 
these  values  in  equation  (i  i),  the  value  of  H^  is  found  to  be 
Om.  16,  which  agrees  somewhat  closely  with  the  actual  meas- 
urements. 

N.B.  —  The  coefficients  a  and  a'  have  not  been  very 
accurately  determined,  but  their  exact  values  are  not  of  great 
importance.  They  are  often  taken  equal  to  unity. 


RUHLMANN'S  LAW.  285 

14.  Longitudinal  Profile  and  Riihlmann's  Law.  —  In  the 

preceding  article,  put  F\i  —  b—j  =  i  —  a—  -j  in  eq.  (7),  then 


=  F.  dh. 


If  the  transverse  profile  has  been  determined,  the  value  of  F 
corresponding  to  the  depth  h  at  any  point  O  can  be  at  once 
found  and,  by  means  of  the  last  equation,  the  surface  profile 
between  the  depths  h  and  H  can  be  easily  plotted. 

Let  Flt  F2,  F3  ,  .  .  .  be  the  values  of  F  at  a  series  of 
points  at  which  the  depths,  differing  successively  by  a  small 
quantity  dh,  are  hv  ,  //2  ,  ^3,  .  .  .  respectively.  Then 


sY  —  Fl  .  dh\     ds2  =  F\  .  dh  ; 


—  F  .  dh; 


and  the  corresponding  distances  s\  ,  s.2  ,  s^,  .  .  '.  of  these  points 
from  O  are 

ds.+ds,  **+<***  .  ,   ds.-l-ds, 

o  i  —  ,    •»  .»  —  •*  i  ~i  —  ,    •>  •>  —  •*  .1  ~T~  i   •    •    • 

^  2  2  2 

EXAMPLE.  A  cut  of  rectangular  section,  with  a  fall  of 
i  in  10,000,  is  10  ft.  wide  and  delivers  40  cu.  ft.  of  water  per 
second.  At  a  certain  point  the  depth  is  increased  to  4  ft.  by  a 
dam.  Assuming  that  the  faces  of  the  cut  are  not  very  smooth 
and  that,  consequently,  .0001  maybe  taken  as  an  approximate 
value  of  b,  then  the  depth,  H,  for  uniform  motion  is  given  by 


=  *y  - 


loH 


10+  2 


or 


80 


and  an  approximate  solution  of  this  equation  is  ff  =  2.9  ft. 

The  following  Table  can  now  be  easily  prepared  for  a  series 
of  depths,  commencing  at  the  dam  and  diminishing  successively 
by  3  ins.,  a  being  unity: 


286 


RUHLM ANN'S   LAW. 


k 

A 

P 

nt 

U 

I  —  a  — 

«* 

F 

ds 

* 

t>i 

4.00 

4° 

18 

af 

T 

.991406 

.000045 

•22031.2 

55o8 

3-75 

37-5 

17-5 

2f 

| 

•991943 

.000041 

24193.7 

6048 

5,77& 

3-5° 

35 

17 

2^y 

! 

.992480 

.0000372 

26679.6 

6670 

12,137 

32-5 

16.5 

i|§ 

§ 

.993017 

.0000335 

29642.3 

7410 

3.00 

30 

10 

If 

8 

•993554 

.00003 

8546 

27,155 

The  tenth  column  gives  the  distances  from  the  dam  of  the 
sections  in  which  the  depths  are  3.75,  3.50,  3.25,  and  3  ft. 

Riihlmann's  formula  for  the  distance  between  two  sections, 
between  which  the  depth  of  the  water  gradually  increases  from 
y  +  tfto  Y+tfis 

•=?{/©-/»)!• 

the  function  frr being  given  by  the  following  table: 


y 

H 

>(*) 

y 

H 

^) 

y 

H 

41) 

O.OI 

O.OO67 

o-3 

.3428 

•4 

2.7264 

O.O2 

0.2444 

0.4 

.5119 

•  5 

2.8337 

0.03 

0.3863 

0.5 

.6611 

.6 

2.9401 

O.O4 

0-4889 

0.6 

v  .7980 

•7 

3-0458 

0.05 

0.5701 

0.7 

.9266 

.8 

3.1508 

O.O6 

0.6376 

0.8 

2.0495 

•9 

3-2553 

O.O7 

0.6958 

0.9 

2.1683 

2.O 

3.3594 

0.08 

0,7472 

I.O 

2.2839 

2-5 

3-8745 

O.O9 

0-7933 

i  .1 

2.3971 

3-o 

4-3843 

0.10 

0.8353 

1.2 

2.5083 

3-5 

4.8910 

0.20 

I.  1361 

i-3 

2.6179 

4.0 

5.3958 

Applying  this  formula  to  the  preceding  example,  in  order 
to  determine  the  distance  between  the  3-  and  4-ft.  depthsf  at 
the  dam 


i.i 


=  -3793, 


CHANNEL   OF  RECTANGULAR  SECTION  AND  SMALL  SLOPE.  287 
and,  by  interpolation, 


=  1.4769. 

At  the  3-ft.  depth 

7f  =  £9  =  -°3448' 

and 

/(£)  =  .4323. 

Hence 

2.9 


.0001 


(1.4769  —  .4323)  =  30,293  feet. 


15.  Channel  of  Rectangular  Section  with  a  nearly  Hori- 
zontal Bed. — In  this  case  i  is  very  small  and  may  be  disre- 
garded in  eq.  (4),  Art.  13,  which  may  therefore  be  written  in 
the  form 

2g  m  ..          2gmdu 

as  = -p-  ~^ah  —  a—.- -. 

/  »2  /  g  u 

But  xhu  =  Q  =  a  constant,  and  therefore  h  .  du  -f-  u  .  dh  =  o. 
Also, 

xh 


Hence 


x*     h*  .  dh          ax      dh 

~  ~    i.  /T.)  7~H i        -,  /.N    ~"  ~     i  „  i       " 


Integrating, 

'is' 


-~   log,  (X  +  2k) 


c  being  a  constant  of  integration. 


288  CHANNEL   OF  GREAT   WIDTH  AS   COMPARED   WITH  DEPTH. 

Hence  the  distance   sl  —  s2  between  two  points  at  which 
the  depths  are  h^  and  k.2  (<  //J  is  given  by 


X*        C^ 

•*  I  *^  *>  ~~r     I^d         / 


X -\-2ll 

The  last  term  is  usually  very  small  and  may  be  disregarded 
without  appreciable  error,  and  therefore 

^     r**  h\ih 


a  formula  by  means  of  which  the  discharge  may  be  found. 

16.  Channel   of  Great  Width    as    compared  with    the 
Depth.  —  In  this  case 

A  =  xh     and     P  —  x,  approximately. 
Therefore 

;;/  —  h     and     M—H. 
Also, 


Hence,  eq.  (7),  Art.   13,  may  be  written  in  the  form 

ai 


\          a 

l   "ft  ~ 

l       ~  3  " 


ds        \H 


dh 


Take  s  =  —  JT~~  —  fj>  ?  being  the  rise  or  fall  above  or 
below  the  surface  of  uniform  motion.       Then  dh  =  H  .  dz,  and 

ai 

ids  ~ 


_       _ 
Hdz~         ""^8-l*. 


CHANNEL   OF  GREAT   WIDTH  AS   COMPARED   WITH  DEPTH.  289 

Integrating, 


being  a  constant  of  integration. 
This  equation  may  be  written 


2s 
Tl 


7          / 

and  between  any  two  points        and      -, 


,  .  (3) 


j/        change  in  depth 
argument  being        =    —.- 


7/z  the  case  of  a  dam  built  across  a  channel  in  which  the 
water  had  previously  flowed  with  a  uniform  motion,  Case  I, 
Art.  13,  in  the  limit, 

si  =  h  —  zH  =  oo  , 
and  therefore,  by  eqs.   I  and  2, 

<£(*)  +  <:  ==  O  =  g  log,  I  +   i7--tan"I°0  +  r=-^-f  +  '» 

and 

<:  —  —  .9069. 

The  following  Table,  calculated  by  Tutton,  gives  the  value 
of  the  backwater  function,  <t>(z),  in  the  case  of  a  dam  : 


290 


BACKWATER  FUNCTION. 


y 
H 

*(*) 

y 

H 

*(*) 

y 

H 

*(*) 

y 

H 

*(*) 

o.ooo 

00 

.072 

.7812 

.285 

.3860 

.68 

•1945 

.001 

2.1837 

.074 

.7727 

.290 

.3816 

.69 

.1918 

.002 

.9530 

.076 

.7644 

.295 

•3773 

.70 

.1892 

.003 

.818.1 

.078 

•7564 

.300 

•3730 

•7i 

.1867 

.004 

.7225 

.080 

.7486 

.305 

.3689 

.72 

.1843 

-005 

.6485 

.082 

.7410 

•  310 

.3649 

•73 

.1819 

.006 

.5881 

.084 

.7336 

•315 

.3609 

•74 

•1795 

.007 

•5379 

.086 

.7264 

'.320 

•3570 

•75 

.1772 

.008 

.4928 

.088 

.7194 

.325 

•3532 

.76 

.1749 

.009 

•4539 

.090 

•7125 

•330 

•3495 

•77 

.1727 

-OIO 

.4191 

.092 

.7058 

•335 

.3458 

.78 

•1705 

.Oil 

.3877 

.094 

•6993 

•340 

.3422 

•79 

.1684 

.012 

.3586 

.096 

.6929 

•345 

•3387 

.80 

.1663 

.013 

.3327 

.098 

.6866 

•350 

•3352 

.81 

.1642 

.014 

.3082 

.  IOO 

.6805 

•  355 

.33i8 

.82 

.1622 

.015 

-2855 

.105 

.6658 

.360 

•3285 

•83 

.  1602 

-Ol6 

.2644 

.  no 

.6518 

.365 

•3252 

.84 

•1583 

.017 

.2446 

•IIS 

.6387 

•370 

.3220 

•85 

.1564 

.018 

.2258 

.  1  20 

.6260 

•375 

.3189 

.86 

.1546 

.Dig 

.2081 

.125 

.6139 

.380 

•3158 

•87 

•  1528 

.020 

•1913 

.130 

.6024 

.385 

.3127 

.88 

•  1510 

.021 

•1754 

•135 

.5913 

•390 

•3097 

.89 

.1492 

.022 

.1602 

.140 

.5807 

•395 

.3068 

.90 

•1475 

.023 

.1457 

•  145 

.5706 

.400 

.3039 

.91 

.1458 

.024 

•1319 

.150 

.5608 

.41 

'  .2982 

.92 

.1441 

.025 

.1186 

•155 

•5514 

.42 

.2928 

•93 

.1425 

.026 

.1059 

.160 

.5423 

•43 

•  2875 

•94 

.1409 

.027 

.0936 

.165 

•5335 

•  44 

.2824 

•95 

•1393 

.028 

.0817 

.170 

•  5251 

•45 

.2774 

.96 

•1377 

.029 

.0704 

•175 

.5169 

.46 

.2726 

.97 

.1362 

.030 

.0595 

.180 

.5090 

•47 

.2680 

.98 

•1347 

.032 

.0387 

.185 

.5014 

.48 

.2634 

•99 

•1332 

-034 

.0191 

.190 

-4939 

^49 

.2590 

1.  00 

.1318 

.036 

.0007 

•  195 

.4867 

•50 

.2548 

1.05 

.1250 

.038 

.9833 

.200 

.4798 

•5i 

.2506 

I.  10 

.1187 

.040 

.9669 

.205 

•4730 

•52. 

.2465 

•15 

.1128 

.042 

•  9512 

.210 

.4664 

•53 

.2426 

.20 

.1074 

.044 

.9364 

.215 

.4600 

•54 

.2388 

•25 

.1024 

.046 

.9223 

.220 

.4538 

•55 

•2351 

•30 

.0979 

.048 

.9087 

-225 

•  4478 

•56 

.2314 

•35 

.0936 

.050 

•8957 

.230 

.4419 

•57 

.22-79 

.40 

.0894 

.052 

•8833 

•235 

•  4363 

•58 

.2245 

•45 

.0856 

•054 

.8714 

.240 

.4306 

•59 

.2212 

•50 

.0821 

.056 

.8599 

-245 

•  4251 

.60 

.2179 

•55 

.0788 

•058 

.8488 

.250 

.4198 

.61 

.2147 

.60 

.0758 

.060 

.8382 

•255 

.4145 

.62 

.2Il6 

•65 

.0728 

.062 

.8279 

.260 

.4096 

•63 

.2086 

.70 

.0700 

.064 

.8179 

•  265 

.4046 

.64 

.2056 

•75 

.0674 

.066 

.8083 

.270 

.3998 

•65 

.2O27 

i.  80 

.0650 

.068 

.7990 

•275 

•3951 

.66 

.1999 

1.85 

.0626 

,O7O 

.7899 

.280 

•3905 

.67 

.1972 

1.90 

.0604 

BACKWATER  FUNCTION. 


291 


y 
H* 

*(*) 

y 

H 

*(*) 

y 

H 

*(*) 

y 

H 

<*>(*) 

i-95 

.0584 

3-4 

.0260 

7.0 

.0078 

20.  o 

.0011 

2.OO 

•  0564 

3-5 

.0248 

8.0 

.0062 

25.0 

.0007 

2.  I 

.0527 

3  6 

.0237 

9.0 

.0050 

30  o 

.0005 

2.2 

.0494 

3-8 

.0218 

IO.O 

.0041 

35-o 

.0004 

2-3 

.  0464 

4.0 

.0201 

II.  0 

.0035 

40.0 

.0003 

2.4 

•0437 

4.2 

.01.85 

12.0 

.0030 

45-o 

.0002 

2-5 

.0412 

4-4 

.OI72 

13-0 

.0026 

50.0 

.0002 

2.6 

.0389 

4.6 

.Ol6o 

14.0 

.0022 

99.0 

.0001 

2.7 

.0368 

4-8 

.0149 

15-0 

.0019 

IOO.O 

.0001 

2.3 

•0349 

5-o 

.0139 

16.0 

.0017 

oo 

.0000 

2.9 

•0331 

5-5 

.0118 

17.0 

0016 

3-o 

.0314 

6.0 

.OIOT 

18.0 

.0014 

3-  2 

.0285 

6-5 

.0089 

19.0 

.0013 

NOTE. — The  corresponding  Table,  deduced  by  Bresse,  whose  argument 

H  4-  y  y 

is  —       —  =  I  -f-       i  maY  be  at  once  obtained   from  the  above  by  adding  i 
H  H 

to  Tutton's  argument. 

In  the  case  of  a  fall,  Case  II,  Art.   13,  in  the  limit 

si  =  h  =  zH  —  o, 
and  therefore,  by  eqs.  (i)  and  (2), 


.and 


I        7f 


The  following  Table,  calculated  by  Tutton,  gives  the  value 
of  the  backwater  function,  cf>(s\  in  the  case  of  a  fall : 


292 


BACKWATER  FUNCTION. 


V 

77 

*(«) 

y 

H 

#*) 

1 

y 
I   H 

*(*) 

y 

H 

*(*) 

0. 

00 

.040 

.5448 

.20 

.9504 

•45 

•5753 

.001 

2.7876 

.042 

•5279 

.21 

•9303 

.46 

•5633 

.002 

2.5562 

.044 

•5TI7 

.22 

.9109 

•47 

•  5515 

.003 

2.4207 

.046 

.4962 

•23 

.8922 

.48 

.5398 

.004 

2.3244 

.048 

.4813 

.24 

.8741 

•49 

•  5282 

.005 

2.2497 

.050 

.4670 

•25 

.8566 

•50 

•5167 

.006 

2.1885 

•055 

•4335 

.26 

•8395 

•5i 

•5054 

.007 

2.1368 

.060 

.4027 

.27 

.8229 

•52 

.4941 

.008 

2.0920 

.065 

•3743 

.28 

.8068 

•53 

.4829 

.009 

2.0525 

.070 

•3479 

.29 

.7910 

•54 

•4717 

.010 

2.0171 

•075 

•3231 

•30 

•7756 

•  55 

.4607 

.012 

•9554 

.080 

.2999 

•31 

.7606 

•56 

•4497 

.014 

.9036 

.085 

.2779 

•32 

•7458 

•  57 

.4388 

•  Ol6 

.8584 

.090 

•2571 

•33 

•7313 

•58 

.4279 

.018 

.8185 

•095 

-2372 

•34 

.7172 

•59 

.4171 

•  O2O 

.7827 

.  IOO 

.2185 

•35 

•  7033 

.60 

.4064 

.022 

.7502 

.11 

.1831 

•36 

.6896 

•  65 

•353f> 

.024 

.7206 

.12 

.1504 

•37 

.6762 

.70 

.3019 

.026 

.6936 

•13 

.1201 

•38 

.6629 

•75 

•  2510 

.028 

.6678 

.14 

.0918 

39 

.6499 

.80 

.2004 

.030 

.6441 

•15 

.0651 

.40 

.6371 

.90 

.  IOOI 

.032 

.6219 

.16 

•0399 

.41 

.6244 

I.OO 

.0000- 

•034 

.6010 

-17 

.Ol6o 

.42 

.6119 

.036 

•5813 

.18 

•9931 

•43 

•5995 

.038 

.5626 

.19 

•9713  j 

•44 

•5873 

1 

NOTE. — Bresse  uses  the  same  value  —  .9069  for  the  con- 
stant c  both  for  a  dam  and  for  a  fall.      His  argument  in  the 


latter  case  is 


H-  y 
H 


V 

-~x  and  to  obtain  Bresse 's  Table 
ri 


from  the  above,  the  argument  adopted  by  Tutton  is  subtracted 
from  I,  and  .6046  from  the  value  of  <J>(z). 

y 

Dupuit,  again,  uses  the  argument  ~y,  and  his  Tables  may 

be  obtained  from  those  given  by  Tutton  by  equating  his  back- 
water function  to 


y 

1.4158  -f-  —  —  <f>(z)  for  a  rise, 
ri 


and  to 


2.0204  —ft- 


for  a  fall. 


CHANGE  OF  CROSS-SECTION. 


293 


,   Dupuit  neglects   the    term   -     -   and  includes  in  his  back- 

<s 

water    function,    which    may    be    designated    f(z),    the    term 

/      y  _  y'\ 
jg  —  #'  (  =       rr     1  in  equation  (3),  so  that  his  formula  becomes 


y 

•considering  that  when  -p.  =  2  =  .01,  accurate  measurement  is 

no  longer  possible.      Ruhlmann  gives  the  same  rule. 

17.  Change  of  Section.  —  CASE  I.,  Fig.  160.  A  channel  of 
slope  2,  and  in  which  the  flow  is  steady,  gradually  contracts 
from  a  width  A  A  =  J5  to  a  width  CC  =  £>,  the  surface  of 


FIG.   1 60. 

steady  motion  being  PQ  above  A  A,  and  RS  below  CC.      On 
approaching  A  A   the  surface  gradually  rises  and   reaches  its 
greatest  height    QT '  =  z  above  PQ  at  AA.      This  is  followed 
by  a  gradual  fall  to  the  surface  of  steady  motion  RS  at  CC. 
Let  kl ,  /22  (>  7/j)  be  the  depths  corresponding  to  steady 
motion    above   AA    and   below   BB, 
respectively. 

"  /#! ,  #/2  be  the  mean  hydraulic  depths  above  A  A  and 
below  BBj  respectively. 

and 


u.2  be  the  mean  velocities  of  flow  above  A  A 


below  BB,  respectively. 


294  CHANGE  OF  CROSS-SECTION. 

Then,  disregarding  the  effect  of  surface  resistance  between 
A  A  and  CC, 


or 


If  the  section  is  a  rectangle, 


But 


Therefore 


and 


If  the  width  is  great  as  compared  with  the  depth, 


>   =  ~     and     ;;/2  =  — ,  approximately. 

2  2 


Therefore 


and 


CHANGE  OF  CROSS-SECTION. 


295 


CASE  II.  A  channel  of  slope  i,  in  which  the  flow  is  steady, 
PQRS  being  the  surface  of  steady  motion,  gradually  contracts 
from  a  width  A  A  =  B,  to  a  narrower  width  at  CC.  The 


FIG.  161. 

channel  remains  narrow  for  a  limited  distance  CD  and  then 
gradually  enlarges  to  its  original  size  at  E.  On  approaching 
A  A  the  surface  rises,  attains  its  greatest  height  QT  above  PQ 
at  A,  falls  to  V  at  C,  then  to  a  point  W below  PQRS  at  D, 
and  finally  suddenly  rises  from  W  to  the  surface  of  steady 
motion  at  R. 

Let  z  be  the  depression  of  W  below  PS. 
4 '  B,  Bl  be  the  widths  at  D  and  E. 
"   u,  ui  be  the  mean  velocities  at  D  and  E. 

Then 


z  - 


where  a  may  be  taken  =  I  .  i  . 
If  the  section  is  a  rectangle, 

B(kl  —  z)u  —  Blulhr 
Therefore, 


a  cubic  equation  giving  z. 


296 


CHANGE  OF  CROSS-SECTION. 


The  surface  DE  may  now  be  plotted,  and  QT  may  be 
found  as  in  Case  I. 

These  expressions  also  give,  approximately,  the  depression 
below  the  surface  PQRS  of  steady  motion  when  the  channel 
has  its  section  suddenly  changed  by  such  obstructions  as 
bridge-piers. 


<t         EE  5> 


FIG.   162. 

On  approaching  the  pier  ends  the  water-surface  gradually 
rises  to  the  maximum  height  T  above  PS,  then  falls  to  XV 
below  PS  between  the  piers,  and  finally  rises  again  to  the  sur- 
face of  steady  motion  on  passing  into  the  open  channel. 

Let  B^ ,  B  be  the  distances  between  the  axes  and  the 
inner  faces  of  the  piers. 

Let  H  be  the  depth  below  XY. 

Let  z  be  the  fall  from  T  to  X. 

Then,  according  to  Bresse,  the  value  of  z  is  given  by  the 
empirical  formula 


• 


B?(H  4- 

cc  being  a  coefficient  of  contraction  and  having  an  average 
value  of  about  .8.  Also,  Q  is  the  discharge  for  the  width  Bl 
of  the  channel. 

*  This  formula,  although  generally  adopted,  is  open  to  question.    Bresse 
considers  that  an  equally  correct  approximation  is  obtained  at  a  distance 

•  /  /? 

of  2O(Bi  —  B)  from  the  contraction  by  taking  z  —  2oiB  [ — -  — 

V  B 


GAUGING.  297 

18.  Gauging  of  Streams  and  Watercourses. — The  amount 
of* flow  Q  in  cubic  feet  per  second  across  a  transverse  section 
of  A  sq.  ft.  in  area  is  given  by  the  expression 


u  being  the  mean  velocity  of  flow  in  the  section  in  feet  per 
second. 

If  the  longitudinal  profile  and  several  transverse  sections  of 
a  channel  can  be  plotted,  the  volume  of  flow  may  be  calculated 
by  means  of  eq.  (i),  p.  275. 

Let  ?/1 ,  u2 ,  .  .  .  un  be  the  mean  velocities,  Al ,  A2,  .  .  .  An 
the  areas,  and  Pl,  P2,  .  .  .  PH  the  wetted  perimeters  of  n  sec- 
tions of  the  channel  at  the  specified  distances  ^  ,  /2 ,  .  .  .  /w_t 
apart.  Then  z,  the  fall  in  the  free-surface  level,  which  may 
be  found  from  the  longitudinal  profile,  is  given  by  the  equation 

z  =  a'"----1-  +     I    —  ~ds, 
in  which 

/  —  /      /  -u/        f  - mi      h  — T 

iH-   2-1  «-i»    —  -  .wa   -          -^' 

and  a  may  be  taken  =  I .  i . 

But  Aji^  —  Au  =  Q  —  /.  u   -  -  An2tn,   and    m  —p- 

Therefore 


and  Q  can  be  calculated  as  soon  as  the  integration  has  been 
effected,  which  may  be  possible  if  P  and  A  are  known  functions 
of  s.  An  approximate  value  of  the  integral  may  be  found 
graphically  as  follows : 

P 

Plot,  as  ordinates,  the  values  of  -,„  at  the  n  sections,  and 

yi 

join  the  upper  ends  of  those  ordinates.      The  area  between  the 


298 


GAUGING. 


extreme  ordinates,  the  axis,  and  the  line  thus  determined  is 
the  value  required. 


FIG.  163. 

Generally  speaking,  however,  the  above  method  of  gauging 
the  flow  of  a  stream  is  not  very  accurate,  on  account  of  the 
errors  in  the  values  of  P,  A,  and  the  integral.  More  correct 
results  are  obtained  by  determining  the  mean  velocity. 

19.  Determination  of  the  Mean  Velocity  u.  METHOD  I. 
The  most  convenient  method  for  gauging  small 
streams,  canals,  etc.,  is  by  means  of  a  temporarily 
constructed  weir,  which  usually  takes  the  form  of  a 
rectangular  notch.  The  greatest  care  should  be 
exercised  to  insure  that  the  crest  of  the  weir  is  truly 
level  and  properly  formed,  and  that  the  sides  are 
truly  vertical.  The  difference  of  level  between  the 
crest  of  the  weir  and  the  surface  of  the  water  at  a 
point  where  it  has  not  begun  to  slope  down  towards 
the  weir  is  best  estimated  by  means  of  Boyden's 
hook-gauge,  Fig.  164. 

This   gauge   consists   of  a   carefully    graduated 
rod,  or  of  a  rod  with  a  scale  attached,  having  at  the 
lower  end  a  hook  with  a  thin  flat  body  and  a  fine 
point.      The  rod  slides  in  vertical  supports, 
and  a  slow  vertical  movement  is  given  by 
means   of  a   screw   of  fine   pitch.      A   stiff 
vertical    rod,    with    a    sharp  point,    having 
been  placed  at  5   to  8  ft.  from  the  back  of 
the  weir,  with  the  point  on  a  level  with  the 
weir  crest,  water  is  run  into  the  flume  until 
it  rises  slightly  above  the  crest,  producing 
a    capillary  elevation  at    the    point.      The  FIG.  164. 

water  is  then   allowed  to  subside  until  this  elevation  is  barely" 


GAUGING. 


299 


perceptible,  when  the  rod  may  be  removed.  A  hook-gauge  is 
hext  placed  in  the  same  position,  and  the  hook  is  slowly  raised 
until  a  capillary  elevation  is  produced  over  its  point.  The 
hook  is  then  slowly  lowered  until  the  elevation  becomes  almost 
imperceptible,  when  a  reading  is  taken  corresponding  to  the 
level  of  the  crest  of  the  weir.  More  water  now  flows  into  the 
flume  and  over  the  weir.  As  soon  as  the  motion  has  become 
steady,  the  hook  is  raised  and  the  point  adjusted  at  the  surface 
in  the  manner  just  described.  A  second  reading  is  taken  and 
the  difference  between  the  two  readings  is  the  head  of  water 
over  the  crest. 

In  ordinary  light,  differences  of  level  as  small  as  the  one- 
thousandth  of  a  foot  can  be  easily  detected  by  the  hook- 
gauge,  while  with  a  favorable  light  it  is  said  that  an  experi- 
enced observer  can  detect  a  difference  of  two  ten-thousandths 
of  a  foot.  Such  differences,  however,  cannot  be  measured 
under  the  ordinary  conditions  of  practical  work. 

METHOD  II.  A  portion  of  the  stream  which  is  tolerably 
straight  and  of  approximately  uniform  section  is  defined  by 
two  transverse  lines  O^AB,  OfD  at  any  distance  5  ft.  apart. 


FIG.  165. 

The   base-line    O^O2  is   parallel  to   the   thread  EF  of  the 
stream,  and  observers  with  chronometers  and  theodolites  (or 


300  GAUGING. 

sextants)  are  stationed  at  Ol ,  Ov  The  time  T  and  path  EF 
taken  by  a  float  between  AB  and  CD  can  now  be  determined. 
At  the  moment  the  float  leaves  AB  the  observer  at  Ol  signals 
the  observer  at  O2 ,  who  measures  the  angle  O^Of.^  and  each 
marks  the  time.  On  reaching  CD  the  observer  at  (9.,  signals 
the  observer  at  Ol ,  who  measures  the  angle  O^O^F,  and  each 
again  marks  the  time. 

Experience  alone  can  guide  the  observer  in  fixing  the  dis- 
tance 5  between  the  points  of  observation.  It  should  be 
remembered  that  although  the  errors  of  time  observations  are 
diminished  by  increasing  5,  the  errors  due  to  a  deviation  from 
lines  parallel  to  the  thread  of  the  stream  are  increased. 

A  number  of  floats  may  be  sent  along  the  same  path,  and 

their  velocities  \-j?j  are  often  found  to  vary  as  much  as  20  per 

cent  and  even  more. 

Having  thus  found  the  velocities  along  any  required  number 
cf  paths  in  the  width  of  the  stream,  the  mean  velocity  for  the 
whole  width  can  be  at  once  determined. 

Surf  ace -floats  are  small  pieces  of  wood,  cork,  or  balls  of 
wax,  hollow  metal  and  wood,  colored  so  as  to  be  clearly  seen, 
and  ballasted  so  as  to  float  nearly  flush  with  the  water-surface 
and  to  be  little  affected  by  the  wind. 

Sub  surf  ace -floats. — A  subsurface  float  consists  of  a  heavy 
float  with  a  comparatively  large  intercepting  area,  maintained 


FIG.  166  FIG.  167. 

at   any  required    depth   by  means   of  a  very  fine   and  nearly 


GAUGING. 


vertical  cord  attached  to  a  suitable  surface-float  of  minimum 
immersion  and  resistance.  Fig.  166  shows  such  a  combina- 
tion, the  lower  float  consisting  of  two  pieces  of  galvanized  iron 
soldered  together  at  right  angles,  the  upper  float  being  merely 
a  wooden  ball. 

Another  combination  of  a  hollow  metal  ball  with  a  piece 
of  cork  is  shown  by  Fig.  167. 

The  motion  of  the  combination  is  sensibly  the  same  as  that 
of  the  submerged  float,  and  gives  the  velocity  at  the  depth  to 
which  the  heavy  float  is  submerged. 

Twin-floats. — Two  equal  and  similar  floats  (Fig.  1 68),  one 
denser  and  the  other  less  dense  than  water,  are 
connected  by  a  fine  cord.      The  velocity  (T^)  of 
the  combination  is  approximately  the   mean  of  rr 
the  surface  velocity  (vs)  and  of  the  velocity  (vd) 
at  the  depth  to  which  the  heavier  float  is  sub- 
Thus 


merged. 


and  therefore 


vf  = 


FIG.  168. 


so  that  vd  can  be  determined  as  soon  as  the  value  of  vt  has 
been  observed  and  the  value  of  vs  found  by  surface-floats. 

Velocity-rod. — This  is  a  hollow  cylindrical  rod  of  adjustable 
length  and  ballasted  so  as  to  float  nearly 
vertical.  It  sinks  almost  to  the  bed  of 
the  stream,  and  its  velocity  (?'„,)  is  ap- 
proximately the  mean  velocity  for  the 
whole  depth. 

Francis  gives  the  following  empirical 
formula  connecting  the  mean  velocity  (vm) 
with  the  observed  velocity  (vr)  of  the  rod : 


FIG.  169. 


OI2   —  .  I  J6. 


V 


w 


302 


GAUGING. 


d  being  depth  of  stream,  and  d'  the  depth  of  water  below 
bottom  of  rod ;  but  d'  should  not  exceed  about  one  fourth  of  d. 
METHOD  III.  Pitot  Tube  and  Darcy  Gauge. — A  Pitot  tube 
(Figs.  170  to  172)  in  its  simplest  form  is  a  glass  tube  with  a 
right-angled  bend.  When  the  tube  is  plunged  vertically  into  the 
stream  to  any  required  depth  z  below  the  free  surface,  with  its 
mouth  pointing  up-stream  and  normal  to  the  direction  of  flow, 


FIG.  170.  FIG.  171.  FIG.  172 

the  water  rises  in  the  tube  to  a  height  h  above  the  outside  surface, 
and  the  weight  of  the  column  of  water,  .7  -+-  //  high,  is  balanced 
by  the  impact  of  the  stream  on  the  mouth.  Hence  (Chap.  V) 


wA( 


—  wAz 


kwA  —  , 


and  therefore 


A  being  the  sectional  area  of  the  tube,  21  the  velocity  of  flow 
at  the  given  depth,  and  k  a  coefficient  to  be  determined  by 
experiment. 

A  mean  value  of  k  is  1.19.  With  a  funnel-mouth  or  a 
bell-mouth  Pitot  found  k  to  be  1.5.  This  form  of  mouth, 
however,  interferes  with  the  stream-lines,  and  the  velocity  in 
front  of  the  mouth  is  probably  a  little  different  from  that  in  the 
unobstructed  stream. 

The  advantages  of  tubes  of  small  section  are  that  the  dis- 
turbance of  the  stream-lines  is  diminished  and  the  oscillations 


GAUGING. 


303 


of  the  column  of  water  are 
checked.  Darcy  found  by 
careful  measurement  that 
the  difference  of  level  be- 
tween the  surfaces  of  the 
water-column  in  a  tube  of 
small  section  placed  as  in 
Fig.  170,  and  of  the  water- 
column  placed  as  in  Fig. 
i  71  with  its  mouth  parallel 
to  the  direction  of  flow,  is 
almost  exactly  equal  to 


When  the  tube  is 
placed  as  in  Fig.  172  with 
its  mouth  pointing  down- 
stream and  normal  to  the 
direction  of  flow,  the  level 
of  the  surface  of  the  water 
in  the  tube  is  at  a  depth 
h'  below  the  outside  sur- 
face, and 

*'  =  *'— 

ft/       -  —    fa  . 

2^ 

where  k'  is  a  coefficient 
to  be  determined  by  ex- 
periment and  a  little  less 
than  unity. 

In  this  case  the  tube 
again  obstructs  the  stream- 
lines. Pitot's  tube  does 
not  give  measurable  indi- 
cations of  very  low  veloci-  ' 
ties.  A  serious  objection 


FIG.   173. 


3°4  GAUGING. 

to  the  simple  Pitot  tube  is  the  difficulty  of  obtaining  accurate 
readings  near  the  surface  of  the  stream.  This  objection  is 
removed  in  the  case  of  Darcy's  gauge,  shown  in  the  accom- 
panying sketch,  Fig.  173. 

A  and  B  are  the  water-inlets  ;  C  and  D  are  two  double 
tubes;  E  is  a  brass  tube  containing  two  glass  pipes  which 
communicate  at  the  bottom  with  the  water-inlets  and  at  the 
top  with  each  other,  and  with  a  pump  F  by  which  the  air  can 
be  drawn  out  of  the  glass  pipes,  thus  allowing  the  water  to  rise 
in  them  to  any  convenient  height. 

Thus  Darcy's  gauge  really  consists  of  two  Pitot  tubes  con- 
nected by  a  bent  tube  at  the  top  and  having  their  mouths  at 
right  angles  or  pointing  in  opposite  directions.  If  Ji  is  the 
difference  of  level  between  the  water-surfaces  in  the  tubes 
when  the  mouths  are  at  right  angles,"  then 


and    Darcy's    experiments  indicate  that  k  does  not    sensibly 
differ  from  unity. 

When  the  mouths  point  in  opposite  directions,  let  /?,  ,  /i2 
be  the  differences  of  level  between  the  stream-surface  and  the 
surfaces  of  the  water  in  the  tube  pointing  up-stream  and  the 
tube  pointing  down-stream,  respectively.  Then 


and  therefore 

h^  -\-  h2  —  (kl  -{- 

-i£- 

K-  , 

where  k  =  k^  -)-  k2. 


GAUGING.  30$ 

k  having  been  determined  experimentally  once  for  all,  the 
difference  of  level  (=  hl-\-  //2)  between  the  columns  for  any 
given  case  can  be  measured  on  the  gauge  and  the  value  of  u 
can  then  be  found. 

A  cock  may  be  inserted  in  the  bend  connecting  the  two 
tubes,  and  through  this  cock  air  may  be  exhausted  and  a  par- 
tial vacuum  created  in  the  upper  portion  of  the  gauge.  The 
water-columns  will  thus  rise  to  higher  levels,  but  the  difference 
between  them  will  remain  constant.  Thus  the  surface  of  the 
column  in  the  down-stream  tube  may  be  brought  above  the 
level  of  the  outside  surface,  and  the  reading  is  then  easily 
made. 

Sometimes  the  gauge  is  furnished  with  cocks  at  the  lower 
parts  of  the  tubes,  and  if  these  cocks  are  closed  when  the 
measurement  is  to  be  made,  the  gauge  may  be  removed  from 
the  stream  for  the  readings  to  be  taken. 

METHOD  IV.  Current-meters. — The  velocity  of  flow  in 
large  streams  and  rivers  is  most  conveniently  and  most 
accurately  ascertained  by  means  of  the  current-meter.  The 
earliest  form  of  meter,  the  Woltmann  mill,  is  merely  a  water- 
mill  with  flat  vanes,  similar  in  theory  and  action  to  the  wind- 
mill. When  the  Woltmann  is  plunged  into  a  current,  a 
counter  registers  the  number  of  revolutions  made  in  a  given 
interval  of  time,  and  the  corresponding  velocity  can  then  be 
determined.  This  form  of  meter  has  gone  out  of  use  and  has 
been  replaced  by  a  variety  of  meters  of  greater  accuracy,  of 
finer  construction,  and  much  better  suited  to  the  work  In  its 
simplest  form  the  present  meter  consists  of  a  screw-propeller 
wheel  (Fig.  1/4),  or  a  wheel  with  three  or  more  vanes  mounted 
on  a  spindle  and  connected  by  a  screw-gearing  with  a  courier 
which  registers  the  number  of  revolutions.  The  met^r  is  put 
in  or  out  of  gear  by  means  of  a  string  or  wire  When  a  cur- 
rent velocity  at  any  given  point  is  to  be  found,  the  reading  of 
the  counter  is  noted,  the  meter  is  sunk  to  the  required  position, 
and  is  then  set  and  kept  in  gear  for  any  specified  interval  0f 


306 


GAUGING. 


time.  At  the  end  of  the  interval  the  meter  is  put  out  of  gear 
and  is  raised  to  the  surface,  when  the  reading  of  the  counter  is 
again  noted.  The  difference  between  the  readings  gives  the 
number  of  revolutions  made  during  the  interval,  and  the 
velocity  is  given  by  an  empirical  formula  connecting  the 
velocity  and  the  number  of  revolutions  in  a  unit  of  time. 

The  vane  V  is  introduced  to  compel  the  meter  to  take  a 
direction  perpendicular  to  that  of  the  stream-lines,  but  this 
may  not  necessarily  be  perpendicular  to  the  axis  of  the  stream. 
The  slight  error  due  to  this  discrepancy  is  usually  disregarded 
in  practice. 

In  order  to  prevent  the  mechanism  of  the  meter  from  being 

FIG.   174. 


FIG.  175. 

injuriously  affected  by  floating  particles  of  detritus,  Revy 
enclosed  the  counter  in  a  brass  box,  Fig.  175,  with  a  glass 
face,  and  filled  the  box  with  pure  water  so  as  to  insure  a  con- 
stant coefficient  of  friction  for  the  parts  which  rub  against  each 
other.  In  the  best  meters,  however,  the  record  of  the  number 
of  revolutions  is  kept  by  means  of  an  electric  circuit,  Fig.  176, 


GAUGING. 


307 


which  is  made  and  broken  once,  or  more  frequently,  each 
resolution,  and  which  actuates  the  recording  apparatus.  The 
time  at  which  an  experiment  begins  and  ends  is  noted,  and  the 
revolutions  made  in  the  interval  are  read  on  the  counter,  which 
may  be  kept  in  a  boat  or  on  the  shore,  as  the  circumstances 
of  the  case  may  require.  The  meter  is  usually  attached  to  a 

FIG.  176. 


FIG.    177. 

suitably  graduated  pole,  so  that  the  depth  of  the  meter  below 
the  water-surface  can  be  directly  read.  In  deep  and  rapid 
water  the  meter  must  be  held  by  a  wire  cord  which  will 
usually  require  to  be  guyed  to  a  forward  line.  The  mean 
velocity  for  the  whole  depth  at  any  point  of  a  stream  may  be 
found  by  moving  the  meter  vertically  down  and  then  up,  at  a 
uniform  rate.  The  mean  of  the  readings  at  the  two  surface" 
positions  and  at  the  bottom  position  will  be  the  number  of 
revolutions  corresponding  to  the  mean  velocity  required.  The* 
mean  velocity  for  the  whole  cross-section  may  also  be  deter- 
mined by  moving  the  meter  uniformly  over  all  parts  of  the 
section. 

The  meter  should  be  rated  both  before  and  after  it  is  used. 
This  is  done  by  driving  the  meter  at  different  uniform  speeds 


308  GAUGING. 

through  still  water.  Experiment  shows  that  the  velocity  if 
and  the  number  of  revolutions  n  are  approximately  connected 
by  the  formula 

v  =  an  -f-  b, 

where  a  and  b  are  coefficients  to  be  determined  by  the  method 
of  least  squares  or  otherwise. 
Exner  gives  the  formula 


VQ  being  the  velocity  at  which  the  meter  just  ceases  to  revolve. 
OTHER   METHODS.-—  Many  other   pieces   of   apparatus  for 
the  measurement  of  current  velocities  have  been  designed. 

Perrodil's  hydrodynamometer,  for  example,  gives  the 
velocity  directly  in  terms  of  the  angle  through  which  a  vertical 
torsion-rod  is  twisted,  and  in  this  respect  is  superior  to  the 
current-meter. 

The    tachometer    or    hydrometric    pendulum    (Fig.     178), 

again,  connects  the  velocity  with 
the  angular    deviation   from   the 
vertical  of  a  heavy  ball  suspended 
by  a  string  in  the  current. 
IX-N~-  v      Hydrometric       and      torsion 

balances  have  also  been  devised, 

but  thev  must  be  regarded  rather 
FIG.  178. 

as    curiosities    than    as   being   of 

any  real  practical  use. 

Having  found  the  maximum  surface  velocity,  vs  ,  at  any 
point  in  a  watercourse,  by  one  of  the  above  methods,  then 
(Art.  10,  p.  259)  the  mean  velocity  of  the  whole  section  is 
given  by  the  empirical  relation 


If  the  transverse  section  of  the  waterway,  at  the  point  in  ques- 


GAUGING,  309 

tion,  is  plotted  and  its  area,  A,  measured,  the  discharge,  Q^ 
may  be  at  once  calculated  by  means  of  the  formula 

* 

<2  =  \Avs. 

Again,  selecting  an  approximately  straight  length  of  channel, 
let  x  be  the  distance  from  the  origin  of  a  particle  in  the  surface 
filament  of  maximum  velocity.  Then  the  velocity  of  this 

particle  is     -,  and  therefore 


Hence 


Qt  =  -  I    Adx, 

SJo 


t  being  the  time  in  which  the  float  passes  over  the  distance  s. 

If  this  distance  is  now  divided  into  n  equal  divisions,  and  if 
A^,  A^  A2,  A3,  .  .  .  An  are  the  areas  of  the  waterway  at  the 
commencement,  at  the  (//  —  I )  intermediate  division  points  and 
at  the  end  of  the  length  s,  then,  by  Simpson's  rule, 


The  integration  may  also  be  at  once  effected  if  A  is  given  as  a 
function  of  x. 

Again,  if  H  is  the  depth  of  steady  motion, 


-- 

b  ~  A* 

and  if  the  width  B  of  the  channel   is  large  as  compared  with 
—  H,  approximately,  and  A  =  BH. 


m 


GAUGING. 


Therefore 


Q  = 


At  any  given  point  in  the  stream  B  may  be  considered  con- 
stant, i  is  also  constant,  and  a  coefficient  m  may  be  substituted 


for  B\     — .      The  actual  depth  //,  which  may  be  read  on  a  fixed 
V   b 

vertical  scale  at  the  point  in  question,  differs  from  H  by  a 
certain  quantity  ;/.  Thus  the  last  equation  may  be  written  in 
the  form 

Q  =    ;,/(//   -f  ;/)t, 

a  convenient  expression  which  is  sometimes  used  to  determine 
the  volume  of  flow  in  wide  rivers.  The  coefficients  m  and  n 
are  constant  at  the  same  point  for  all  depths,  but  vary  from 
point  to  point. 


TABLE    GIVING  THE  VALUES    OF  m  AND    ;/,  THE  UNIT  BEING 
A    METRE    OR    A    FOOT. 


t 

n 

i 

t 

Locality. 

Metres. 

Feet. 

Metres. 

Feet. 

Authority. 

Mantes     bridge      on     the 
Seine 

QC 

*6* 

7 

2  "\ 

Roanne     bridge     on     the 

1  80 

1070 

.25 

.82 

Graeff 

Come  bridge  on  the  Adda 

100-3.2  h. 

594-5.8  h. 

.O 

.O 

Lombardini 

NOTE. — From  an  examination  of  a  large  number  of  gaugings  Bresse 
infers  that  u  =  .8527,  gives  better  average  results  than  u  —  .Svs  (Art.  10  and 
p.  259).  The  latter,  however,  is  equally  safe  unless  it  is  necessary  to  pro- 
vide against  floods.  (Ann.  des  Fonts  et  ChaussJes,  1897.) 


GAUGING. 


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317 


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GAUGING. 


VALUES    OF  b  AND  c,    FOR    THE    SIX  CLASSES    I    TO    VI,    P.   250, 
IN    BAZIN'S    NEW  FORMULA,  bv1  =  mi,  OR  v  =  cVini,  WHERE 

=  87  i/^7/,    OR  =  157.6  t/,,/,    ACCORDING     AS     THE 


(  1  +  -Z=?\  = 
\        Vmt 


UNIT    IS    A    METRE    OR    A    FOOT. 


Value  of  /«, 
the  Unit 
being  a 

Metre   Foot. 

CLASS  I, 
the  Unit  being  a 

CLASS  II, 
the  Unit  being  a 

CLASS  III, 
the  Unit  being  a 

Me 

b 

tre. 

c 

Fo 
b 

ot. 
c 

Me 
b 

tre. 
c 

Fo 
b 

ot. 
c 

Me 
b 

tre. 
c 

F 

b 

oot. 
c 

' 

\ 

•°5 

.16 

.0146 

68.5 

.00445 

124.1 

•  0197 

50-7 

.00600 

91.8 

•0352 

28.4 

.01073 

51-* 

.06 

.20 

•  0143 

69.8 

.00436 

126.4 

.0190 

52-6 

.00579 

95-3 

•Q331 

30.2 

.oioo9 

54-5 

.07 

•23 

.0141 

70.9 

.00430 

128.4 

.0185 

54-2 

.00564 

98.2 

•0315 

3!-7 

.0096° 

56.3- 

.08 

.26 

.0139 

71.8 

.00424 

130.0 

.0180 

55-6 

.00549 

00.7 

.0302 

33-i 

.0092° 

59-9- 

.09 

•30 

.0138 

72-5 

.00421 

131-3 

.0176 

56.7 

.00536 

02.7 

.0291 

34-4 

.oo887 

62.3. 

.  10 

•33 

•0137 

73-1 

.00418 

132.4 

•  0173 

57-7 

.00527 

04-5 

.0282 

35  -S 

.oo856 

64-  a 

.  ii 

•36 

.0136 

73.6 

.00415 

*33-3 

.0170 

58.7 

.00518 

06.7 

.0274!   36.^ 

.00835      66.1 

.  12 

•39 

•  OI35 

74.1 

.00411 

124.2 

.0168 

59-5 

.00512 

08.3 

.0268 

37-4 

-oo8i7|     67.7 

•J3 

•43 

•  0134 

74.6 

.00408 

135-* 

.0166 

60.2 

.  00506 

09.0 

.0262 

38-2 

.oo799 

69.2 

.14 

.46 

•  0133 

75-o 

.00405 

135-8 

.0164 

60.9 

.  00500 

10.3 

.0256 

3Q-o 

.0078° 

70.6 

•15 

•49 

75-3 

" 

T36-3 

.  o  1  63 

61.5 

.00497 

11.4 

.0252 

39-7 

.0076° 

71.9 

.16 

•52 

.0132 

75-6 

.00402 

136.9 

.0161 

62.1 

.00491 

12-5 

.0247 

40.5 

•oo753 

73-4 

-1? 

•56 

" 

75-9 

*k 

'37.4 

.0160 

62.7 

.00488 

13.6 

.0243 

41.2 

.0074! 

74-7 

.18 

•59 

,OT3IJ     76.2 

.00399 

138.0 

.0158 

63.2 

.00482 

14.5 

.0240 

41.  8 

.00732 

75-8 

.19 

.62 

"       1     76.5 

138.5 

-0157 

63-6 

.00479      15.2 

.0236 

42.4 

.00719 

76.9 

.20 

•65 

.0130 

76.7 

.00396 

138.9 

.0156 

64.1 

.00476 

16.1 

•0233 

42.9 

.0071° 

77-7 

.21 

.69 

" 

76.9 

" 

'39-3 

•  0155 

64-5 

.00473 

16.8 

.0230 

43-5 

.0070' 

78.8 

.  22 

•  72 

" 

77-1 

11 

139-6 

.0154 

64  9 

.00470 

17.6 

.0228 

44-o 

.00695 

79-7 

•23 

•75 

.0129      77-3 

.00393 

140.0 

.0153 

65-2 

.00467 

18.1 

.0225 

44-4 

.00686 

80.  i 

•24 

•79 

77-5 

140.4 

65-5 

18.6 

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44-8 

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81.2 

-25 

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"     !   77-6 

" 

M°-5 

.0152 

65-9 

.00463 

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.0221 

45-3 

.  00674 

81.7 

.26 

•85 

"        77-8 

11 

140.9 

.0151 

66.2 

.00460 

19.9 

.0219 

45-7 

.00668 

82.  a 

.27 

.88 

.0128!   78.0 

.00390 

I4'-3 

.0150 

66.5 

•00457 

20.4 

.0217 

46.1 

.00662 

83-5 

.28 

.92 

"        78.1 

J4i   5 

66.8 

21  .O 

.0215 

46.5 

.  00654 

84.2 

.29 

•95 

;;  1  78.3 

" 

141.8 

.0149 

67.0 

.004^54 

21-4 

.0213 

46.9 

.00649 

84.9 

.30 

.98 

78.4 

44 

142.0 

67-3 

21.9 

.0211 

47-3 

.00643 

85-7 

•3* 

.02 

" 

78.5 

" 

142.2 

.0148 

67.6 

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22.5 

.0210 

47-6 

.00639 

86.2 

•S2 

•05 

.0127    78.  b 

-00387 

142.4 

4* 

67.8 

22-9 

.0209 

47  9 

.00637 

86.7 

-33 

.08 

"        78-8 

14 

142.7 

.0147 

68.0 

.00448 

23.2 

.0208    48.2 

.00634 

87.3 

•34 

.     2 

789 

" 

142.9 

68.2 

*  * 

23.5 

.0206 

48.5 

.00628 

87  8 

•35 

.    5 

79.0 

" 

143  -1 

.0146 

68.4 

.00445 

23-9 

.0204 

48.8 

.00622 

88.3. 

•36 

.  8 

" 

79.1 

" 

*43-3 

44 

68.6 

24.2 

.0203 

49-2 

.00619 

80.  r 

•37 

.   i 

.0126 

79.2 

.00384 

M3-5 

.0145 

68.8 

.00441 

24.6 

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49-5 

.00616 

89  6 

.38 

e 

•l 

4* 

69.0 

44 

25.0 

.0201 

49.8 

.00613 

90.2 

•39 

.    8 

" 

79-3 

U 

143.6 

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69.2 

.00438 

25-4 

.0200 

50.1 

.00609 

9°-7 

.40 

•31 

44 

79-4 

" 

143-8 

69.4 

125.7 

•0199 

50.4 

.00606 

91.2 

.41 

34 

" 

79-5 

** 

144.1 

44 

69.6 

'4 

126.1 

44 

50.6 

44 

9T-5 

.42 

•38 

" 

79.6 

^ 

144    2 

.0143 

69.7 

.00436 

126.2 

.0197 

5o.9 

.  00600' 

92.2 

•43 

.41 

44 

79-7 

41 

144.4 

69.9 

126.7 

.0196 

51.  i 

.00597 

92.6 

•44 

•44 

.0125 

%l 

.00381 

*4 

" 

70.1 

" 

I27.O 

•  OI9S 

51-4 

•00595 

93-i 

•45 

•47 

79-8 

4* 

144.5 

..0142 

70.2 

.00433 

127.2 

.0194 

Si  6 

.00591 

93-5 

.46 

•51 

" 

79-9 

44 

144-7 

70.4 

kt 

127.4 

•0193 

51-8 

.00588 

93  •& 

•47 

•54 

" 

fco.o 

14 

144.9 

44 

70.5 

" 

I27.7 

.0192 

52.0 

.00585 

94.2 

.48 

•57 

" 

44 

4  4 

44 

70.6 

41 

127.9 

.0191 

52-3 

.00582 

04  7 

•49 

.61 

" 

80.  i 

44 

145.1 

.0141 

70.8 

.00430 

128.2 

52-5 

44 

95-i 

•5° 

.64 

" 

80.2 

44 

MS  -3 

44 

70-9 

128.4 

.0190 

52-7 

•00579 

95-5 

•55 

.80 

.0124 

80.4 

.00378 

145.6 

.0140- 

7'.  5 

.00427 

129.5 

.0186 

53-7 

.00567 

97-3 

.60 

•97 

80.7 

146.2 

.0139 

72.1 

.00424 

130.6 

.0183 

54-6 

.00558 

08.9 

•65 

•'3 

" 

80.9 

" 

146.6 

.0138 

72.6 

.00421 

131.5 

.0181 

55-4 

.00552 

100.3 

.70 

•3° 

.0123 

81.  i 

•00375 

146.9 

.0137 

73.0 

.00418 

132.2 

.0178 

56.1 

.00543 

101.6 

•75 

.40 

81.3 

'' 

147-3 

.0136 

73-4 

.00415 

132.9 

.0176 

5-5.8 

•00537 

102.9 

.80 

.62 

" 

81.5 

44 

147.6 

73-8 

44 

133-6 

.0174 

57  4 

.00530 

104.0 

.85 

•79 

.0122 

81.7 

.00372 

148.0 

•OI35 

74.1 

.00411 

*34-2 

.0172 

58.0 

.00524 

105.1 

.90 

95 

81.8 

148.2 

.0134 

74-4 

.00408 

134-8 

.0171 

58.6 

.00521 

106.1 

320 


GAUGING. 


VALUES    OF    b   AND    c,   FOR  THE    SIX    CLASSES    I  TO  VI,   P.   250, 
IN   BAZIN'S    NEW    FORMULA,  bit  =  mi,  OR  v  =  c^^mi,  WHERE 


c(\+  -Z=}   =87   Vtni,  OR  =  157.6  Vmi',    ACCORDING    AS    THE 

\        Vml 
UNIT    IS    A    METRE    OR    A    FOOT. 


Value  of  m 

CLASS  I, 

CLASS  II, 

CLASS  III, 

the  Unit 

the  Unit  being  a 

the  Unit  beinga 

t-he  Unit  being  a 

being  a 

Metre. 

Foot. 

Metre. 

Foot. 

Metre. 

Foot. 

Metre 

Foot 

b 

c 

b 

c 

b 

C 

b 

c 

b 

c 

b 

C 

•95 

3.12 

.OT22 

81.9 

.00372 

148.4 

.0134 

74.7    .00408 

135-' 

.0160 

59  i 

.00515 

107.0 

i  .00 

3-28 

44 

82.0 

(4 

148.5 

•0133 

75-o 

.00405 

135-8 

.0168 
o-^ 

59-6 

.00512 

107.9 

1.20 

3.61 
3-94 

.OI2I 

82.4 

.00369 

148.9 
149-3 

.0132 

75-4 
75-9 

.00402 

'36.5 
'37  4 

.016. 

60.5 
61.3 

.0050^ 
.00497 

ID9.6 
III  .O 

I.30 

4.26 

«' 

82.6 

*• 

149.6 

.0131 

76.3 

.00399 

138.2 

.0161 

62  o 

.00491 

112.4 

1.40 

4-59 

t< 

82.8 

" 

150-0 

44 

.0160 

62.6 

.00488 

ri3-4 

I.50 
I.  60 

4.91 

5-25 

.OI2O 

82.9 
83-0 

.00366 

150-2 
150.4 

.0130 

76.9 
77.2 

.00396 

J39-3 
139.  « 

.0158 
.oiJ7 

63.2 
6?,.  8 

.00482 
.00479 

114-5 
115-6 

.        1.70 

5.58 

" 

83.1 

'' 

150.5 

.0129 

77-5 

.00393 

140.3 

.0156 

64-3 

.00476 

116.5 

I.  80 

5.91 

" 

83.* 

" 

150.7 

77-7 

140.7 

.0154 

64.8 

.00470 

117.4 

I.QO 

6.24 

44 

83.3 

44 

150.9 

.0128 

77-9 

.00390 

141.1 

•015.^ 

65.2 

.00466 

118.1 

2.00 

6.57 

" 

83-4 

44 

151-1 

44 

78.1 

141   5 

.0152 

6s.  6 

.00463 

118.9 

2.20 

7.22 

" 

83.6 

14 

151-4 

.0127 

78.5 

.00387 

142.2 

.0151 

66.4 

.00460 

120.3 

2.40 

7.87 

.OlTg 

83.7 

.00^63 

151-6 

44 

78.8 

.< 

142.8 

.0149 

67.1 

.00454 

121.5 

2.00 
2.80 

8-53 
9.19 

t< 

83.8 
83-9 

!< 

151.8 
152.0 

.0126 

79-i 
79  4 

.00384 

143-3 
143-9 

.0148 
.0147 

67.7 
68.2 

•00451 
.00448 

122.6 

I23-5 

3.00 

9.84 

" 

84.0 

" 

152.2 

44 

79.6 

,44-2 

.0146 

68.7 

.00445 

124-4 

3.20 

10.50 

44 

84.1 

u 

152.4 

.0125 

79.8     .00381 

144.6 

.0145 

69.2 

.00442 

I25-3 

3-4° 

11.15 

" 

84.2 

41 

i52-5 

>4 

80.0 

144.9 

.0144 

^9.6 

.00439 

126.0 

3-6o 

11.81 

44 

84-3 

" 

152-7 

44 

80.2 

44 

145-3 

.0143 

70.0 

.00436 

126.8 

3-8o 

12.47 

0118 

84.4 

i52-9 

.0124 

80.4 

.00378 

145.6 

.0142 

70.4 

.00433 

53:; 

4  .00 
4-5° 

13.12 
14.76 

84.6 

T53-3 

80.9 

44 

146.5 

.0140 

7l-5 

.00430 
.00427 

129.5 

5.00 

16.4 

" 

84.7 

44 

153-4 

.0123 

81.2 

•°04375 

147.0 

•  onq 

72.1 

.00424 

130.6 

5-5° 

18.04 

" 

84.8 

" 

153-6 

.0123 

8.  .4 

146.6 

.0138 

72.7 

.00421 

J3i-7 

6.00 

19.69 

" 

84-9 

" 

«53.8 

4* 

81.6 

41 

147  8 

•  0137 

73-2 

.00418 

132.6 

6.50 

2i-33 

" 

85.0 

11 

154.0 

.0122 

81.8 

.00372 

148.2 

.0136 

73-7 

.00415 

133-5 

7.00 

22.96 

'* 

** 

44 

" 

82.0 

148.5 

•0135 

74.1 

.  OO4  I  2 

T34.2 

7-5o 

23.61 

" 

85.1 

" 

154-2 

44 

82.2 

44 

148.9 

•  0134 

74-5 

.00408 

134-9 

8.00 

26.25 

" 

85.2 

" 

1  54  «3 

44 

82.3 

•4 

149.1 

44 

74.8 

44 

I35-5 

8.50 

27.89 

.0117 

.00356 

.0121 

82.4 

.00369 

J49-3 

.0133 

75-i 

.00405 

136.0 

9.00 

29-53 

44 

85-3 

u 

154-5 

41 

82.6 

M 

149.6 

75-4 

44 

*36.  5 

9-5° 

3i.i7 

" 

*fc 

" 

4* 

82.7 

44 

149.8 

.0132 

75-7 

.00402 

i37-o 

o  oo 

32  81 

44 

*' 

44 

** 

44 

82.8 

44 

150.0 

44 

7S-9 

44 

'37-4 

I.  00 

36.09 

" 

85-4 

* 

154-7 

14 

83-0 

" 

150.4 

.0131 

76.4 

.00399 

138.4 

2.OO 

39-37 

44 

85.5 

4 

154.9 

.Ot20 

83.1 

.00366 

150.5 

.0130 

76.8 

.00396 

139.  i 

3  -co 

42-65 

11 

** 

* 

" 

83-3 

44 

150.9 

44 

77.1 

T39-7 

4.00 

45-93 

" 

85.6 

1 

i55-o 

" 

83-4 

44 

151-1 

.0129 

77-4 

.00393 

140.2 

5.00 

49.21 

" 

fc* 

* 

4i 

44 

83-5 

44 

i5i-3 

44 

77-7 

l* 

140.7 

6.00 

52-49 

44 

B5.7 

1 

155-2 

44 

83.6 

44 

i5T-4 

.0128 

78.0 

.00390 

141.2 

7.00 

55.77 

" 

4 

.0119 

83-7 

.00363 

151.6 

44 

78.3 

141.8 

8.00 

59  -°6 

4i 

44 

* 

** 

4% 

83.8 

151.8 

.0127 

78.5 

.00387 

142.2 

9.00 

?-£4 

it 

85.8 

* 

^55-4 

44 

83-9 

44 

152.0 

44 

78.7 

i  t 

142.6 

20.00 

65.62 

152.2 

78.8 

142.8 

GAUGING. 


VALUES   OF   b  AND    <r,    FOR  THE    SIX    CLASSES    I    TO  VI,  P.  250, 
IN  BAZIN'S  NEW  FORMULA,  t>v*  =  mi,  OR   v  -  fVmi,  WHERE 


C\l  +  -~\   =  S7Vmt't   OR  =  157.6 
\        Vm) 
UNIT    IS    A    METRE    OR    A    FOOT. 


ACCORDING     AS     THE 


Value  of  tn, 

CLASS  IV, 

CLASS  V, 

CLASS  VI, 

the  Unit 
being  a 

the  Unit  being  a 

the  Unit  being  a 

the  Unit  being  a 

Metre. 

Foot. 

Metre. 

Foot. 

Metre. 

Foot. 

Metre 

Fogt. 

3 

C 

b 

C 

b 

c 

b 

c 

b 

c 

b 

' 

•  05 
.06 

.16 

•0552 

18.1 

.01682 

32-8 

.0784 

12.8 

.02390 

23.2 

•  1015 

9-9 

I  .03094 
.02856 

17.9 

.07 

•  23 

.0484 

20  6 

•01475 

37-3 

.0680 

14.7 

.02073 

26.6 

•°937 

.o876 

Jo.  7 

11.4 

.02670 

19  •  4 

20.6 

.08 

.26 

.0461 

21.7 

.01405 

39-3 

.0644 

15-5 

.0196: 

28.1 

.0827 

12.  I 

.02521 

21.9 

.09 

•30 

.0441 

22.7 

•01345 

41.1 

.0613 

16.3 

.01868 

29-5 

.0786 

12.7 

.02396 

23-0 

.  IO 

•33 

.0424 

23.6 

.OI2Q2 

42.7 

.0588 

17.0 

.01792 

30.8 

•075* 

13-3 

.02289 

24.1 

.11 

•36 

.0410 

24.4 

.01250 

44.2 

.0566 

17.7 

.01725 

32.1 

.0722 

13-9 

.02201 

25.2 

.  12 

•39 

•0397 

25-2 

.OI2IO 

45-6 

.0547 

18.3 

.01667 

33-2 

.0696 

14.4 

.O2I2I 

26.1 

•J3 

•43 

.0386 

25.9 

.01177 

46.9 

.0530 

18.9 

.01615 

34-2 

.0673 

14.9 

.02055 

27.0 

.14 

.46 

0376 

26.7 

01146 

47-5 

•0515 

19.4 

.01570 

35-i 

.0653 

*5-3 

.01990 

27-7 

.15 

•49 

.0367 

27.2 

.01119 

49-3 

.0501 

19.9 

.01544 

36.0 

.0625 

15-8 

.01936 

28.6 

.16 

•52 

•°359 

27.8 

.01094 

5°-3 

.0489 

20.4 

.01490 

36-9 

.0618 

16.2 

.01884 

29-3 

•17 

•56 

.0352 

28.4 

.0107; 

5i-4 

.0478 

20.9 

•oi457 

37-9 

.0603 

16.6 

.01839 

30.1 

.18 

•59 

•°345 

29.0 

.01052 

52.5 

.0467 

21.4 

.0142; 

38.8 

.0589 

17.0 

-01795 

30.8 

.19 

.62 

•°339 

29-5 

.01033 

53-4 

.0458 

21.8 

.01396 

39-5 

•°577 

17-3 

-01759 

3T-4 

.20 

•65 

°334 

30.0 

.oioif 

54-3 

.0449 

22.3 

.01369 

40.4 

•0565 

17.7 

.01722 

32.1 

.21 

.69 

.0328 

3°-5 

.00997 

55-2 

.0441 

22.7 

•01345 

41.1 

•0554 

18.1 

.01689 

32.8 

.22 

.72 

•0323 

3°-9 

.00985 

56.0 

•0434 

23-1 

.01324 

41.8 

•0544 

18.4  .01658 

33-7 

•23 

•75 

.0319 

3i-4 

.00972 

56-9 

.0427 

23-4 

.01302 

42.4 

•0535 

18.7  .01631 

33-9 

.24 

•79 

•0315 

3'.8 

.00960 

57-6 

.0420 

23  8 

.01280 

43-i 

.0526 

19.0  .01603 

34-4 

•25 
.26 

.82 

Re 

.0310 

32.2 

72  6 

.00945 

58.3 

.0414 
.0408 

24.2 

.01262 

43.8 

.0518 

19.3  .01579 

34-9 

.27 

•  °5 
.88 

.0307 
.0303 

3.2  .0 
33-o 

.00930 

.00924 

59  •*•* 

59.8 

.0403 

24  •  5 
84.8 

.01229 

44-9 

.0502 

19.9  .01528 

35-5 
36.0 

.28 

.92 

.0300 

33-4 

.00915 

60.5 

.0397 

25.2 

.OI2IO 

45-6 

.0495 

2O.  2   .01507 

36.5 

.29 

•95 

.0297 

33  7 

.0090; 

61  .0 

•0393 

25-5 

.01198 

46.2 

.0489 

20.5   .01489 

37-1 

.30 

.98 

.0293 

34-1 

.0089; 

61.8 

.0388 

25-8 

.01183 

46.8 

.0482 

2O-7  ;  .01468 

37-4 

•31 

1.02 

.0291 

34-3 

.00887 

62.1 

.0383 

26.J 

.01167 

47-4 

.0476 

21.0    .01450 

38.0 

•32 

•05 

.0288 

34.7  .00878 

62.9 

•0379 

26.4 

•OU55 

48.0 

.0471 

21.2  ;  .01435 

38.4 

•33 

.08 

.0285 

35  .  i  j  .  00869 

63.6 

•0375 

26.7 

.01143 

48.6 

.0465 

21.5  .01417 

38.9 

•34 

.12 

.0283 

35.41.00863 

64.1 

•0371 

26.9 

.01131 

48.7 

.0460 

21.7  \  .01402 

39.3 

•35 

•15 

.0280 

35-7  1-00853 

64-7 

.0368 

27.2 

.O1I22 

49-3 

•°455 

22.  O 

.01387 

39-9 

•36 

.18 

.0278  36.0 

.00847 

65.2 

.0364 

27.5 

.OHIO 

49-9 

.0450 

22.2 

.01372 

40.1 

•37 

.21 

.0276  36.3 

.00841 

65-7 

.0361 

27.7 

.01100 

50.2 

.0446 

22.4 

.01360 

40.6 

•38 

•25 

.0274  36.6 

.00835 

66.3 

•0357 

28.0 

.01088 

50.8 

-0441 

22.7 

.01344 

41.1 

•39 

.28 

.0272 

36-8 

00829 

66.6 

•0354 

28.2 

.01079 

Si-* 

•0437 

22-9  1  .01331 

41.4 

.40 

•31 

.0270 

37-1 

.00823 

67.! 

•0351 

28.5 

.01070 

S1-? 

•°433 

23.1   .01319 

41.8 

.41 

•34 

.0268  37.4 

.00817 

67.7 

•  0349 

28.7 

.01064 

52.0 

.0429 

23-3   -01307 

42.2 

.42 

•38 

.0266  !  37.6 

.00811 

68.1 

.0346 

28.9 

01055 

52-3 

.0426 

23-5 

.  o  i  298 

42.5 

•43 

.41 

.0264  1  37-9 

.00805 

68.6 

•  0343 

29.2 

.01046 

52.9 

.0422 

23-7 

.01286 

42.9 

•44 

•44 

0262 

38-1 

00799 

69.1 

.0340 

29  4 

.01036 

53-2 

.0418 

23-9 

.01274 

43-3 

•45 

•47 

.0261 

38-4 

.00796 

69.6 

.0338 

29.6 

.01030 

53-6 

.0415 

24.1 

.OI265 

43-7 

.46 

•5i 

.0259 

38-6 

.00789 

69.9 

•0335 

29.8 

.01021 

54-o 

.0412 

24.3 

.01256 

44-o 

•47 

•54 

.0258 

38.8 

.00786 

7°-5 

•°333 

30.0 

.01015 

54-3 

.0409 

24-5 

.01247 

44-4 

.48 

•57 

.0256 

39  i 

.00780 

70  8 

•  0331 

30.2 

.01009 

54-7 

.0405 

24.7 

.01234 

44-7 

•49 

.61 

.0255 

39-3 

.00777 

71.2 

.0329 

3°-4 

.01003 

55-o 

.0403 

24.8 

.OI228 

44  '9 

•50 

.64 

0253 

39-5 

.00771 

71  5 

•  0326 

30.6 

.00994 

55-4 

.0400 

25.0 

.01189 

45-3 

•55 

.80 

.0247 

4°-5 

00753 

73-4 

•  0317 

31-6 

.00966 

57-2 

.0386 

25-9 

.01177 

46.9 

.60 

•97 

.0241  j  41.4 

.00734 

75-o 

.0308 

32.5 

.00939 

58-9 

•0375 

26.7 

.O1143 

48.3 

.65 

•J3 

.0236]  42.3 

.00719 

76.6 

.0300 

33-3 

00914 

60.3 

.0365 

27.4 

.OIII2 

49  6 

.70 

•3° 

.0232 

43-t 

00707 

78.1 

.0294 

34-1 

.00896 

61.8 

.0356 

28.1 

.01085 

5°-9 

•75 

.46 

.0228 

43-9 

.00695 

79-5 

.0288 

34-8 

.00878 

63.0 

.0347 

28.8 

.OIO58 

52-1 

.80 

.62 

.0224 

44-6 

.00683 

80.8 

.0282 

.00856 

64-3 

.0340 

29.4 

.01036 

53-2 

•85 

•79 

.0221 

45-2 

.00674 

81.9 

.0277 

36.1 

00841 

65.4 

•°333 

3O.O 

.01015 

54-3 

322 


GAUGING. 


VALUES    OF   b   AND    c,   FOR    THE    SIX    CLASSES    I    TO  VI,  P.  250, 
IN    BAZIN'S    NEW  FORMULA,  bv*  =  mi,  OR  v  =  cVmi,  WHERE 

7,  OR   =    157.6  Vrni,    ACCORDING     AS    THE 
UNIT  IS  A  METRE  OR  A  FOOT. 


~==    ==  &J 
Vml 


Value  of  /«, 

CLASS  IV, 

CLASS  V, 

CLASS  VI, 

the  Unit 

the  Unit  being  a 

the  Unit  being  a 

the  Unit  being  a 

being-  a 

Mef*»- 

Foot 

Foot. 

Metr^ 

Fr. 

Metre 

Foot. 

b 

e 

b 

e 

b 

c 

b 

c 

b 

c     b 

C 

.90 

2-95 

.0218 

45-9 

.00663 

83.1 

.0273 

36-7 

.00829 

66.5  .0327 

30.6 

.00997 

55-4 

•95 

3.12 

.0215 

46.5 

.00654 

84.2 

.0267 

37-3 

00821 

67.6 

.0321 

31-1 

.00979 

56.3 

.00 

3-28 

.0213  47.0 

.00649 

85.1 

.0265 

37-8 

.00795 

68.5 

.0316 

31.6 

.00963 

57-2 

.  10 

.0208  48.0 

.00634 

86.9 

.0258 

38.8 

00786 

70.3 

.0307 

32.6 

.00936 

59-o 

.20 

3-94 

.0204 

48.9 

.00622 

88.6 

.0251 

39-7 

00765 

71.9 

.0299 

33-5 

.00911 

60.7 

-3° 

4.26 

.0201 

49-8 

.00613 

00.2 

.0246 

40.6 

00749 

73-5 

.0291 

34-3 

.00887 

62.1 

.40 

4-59 

.0198 

50.6 

.00604 

91.6 

.0241 

41.4 

00734 

74-9 

.0285 

35  l 

.00869 

63.6 

•5° 

4.91 

.0195 

SI-3 

.00595 

92-9 

.0237 

42.2 

.00722 

76.3 

.0279 

35-8 

.00850 

64.8 

.60 

5-25 

.0192 

52.0 

-00585 

94.2 

•0233 

42.9 

.00710 

77-7 

.0274 

36-5 

.00835 

66.2 

•7° 

5.58 

.0190 

52.6 

.00579 

95-3 

.0230 

43-6 

.00701 

79.0 

.0269 

37-1 

.  00820 

67.2 

.80 

5-91 

.0188 

53-2 

•c°573 

96.3 

.0226 

44-2 

00689 

80.  i 

.0265 

37-7 

.00808 

68.3 

.90 

6.24 

.0186 

53-8 

.00567 

97-4 

.0223 

44-8 

00680 

81.2 

.0261 

38.3 

.00796 

69.4 

.00 

6.57 

.0184 

54-3 

.00561 

08.4 

.0221 

45-3 

.00674 

81.7 

.0257 

38-9 

.00784 

.20 

7.22 

.0181 

55-3 

.00552 

IOO.2 

.O2l6 

46-4 

.00659 

84.0  .0251 

39-9 

.00765 

72-3 

.40 

7.87 

.0178 

56.2 

.00543 

101.  I 

.0212 

47-3 

.00646 

85.7  .0245 

40.8 

.00747 

73-9 

60 

8.53 

•0175 

.00534 

103.  I 

.0208 

48.1 

.00634 

87.1 

.0240 

41.7 

.00732 

75-1 

.80 

9.19 

57-7 

.00528 

104.4 

.0204 

48-9 

.00622 

88.5 

.0235 

42-5 

.00717 

77  o 

3.00 

9.84 

.0171 

58.3 

.00521 

105.6 

.O2OI 

49-7 

.00613 

89.9 

,0231 

43-3 

.00705 

78.4 

3.20 

10.50 

.0170 

58-9 

.00518 

106.7 

.0199 

50.4 

.  00607 

91.2 

.0227 

44.0 

.00693 

79-7 

3-4° 

11.15 

.0168 

59-5 

.00512 

107.8 

.0196 

51.0 

.00598 

92.3 

.0224 

44.6 

.00683 

80  8 

3-6o 

11.81 

.0167 

60.  t 

.00509 

108.9 

.0194 

.00592 

93-5 

.0221 

45-2 

.00674 

81  .9 

12.47 

.0165 

60.6 

.00503 

109.8 

.0192 

52.2 

.00585 

94.6 

.0218 

45-8 

.00665 

83.0 

4.00 

13.12 

.0164 

61  .0 

.00500 

110.5 

.Oigo 

52.7 

-00579 

95-5 

.0216 

46.4 

.00657 

84.0 

4-50 

14.76 

.0161 

62.1 

.00491 

112-5 

.Ol86   53.9 

.00567 

97-6 

.0210 

47-6 

.00639 

86.2 

16.40 

.0159 

63.0 

.00485 

II4.I 

.0182   55.0 

•00555 

99.6 

.0205 

48.8 

.  00624 

88.3 

5-50 

18.04 

.0157 

63.8 

.00479 

"5-5 

.0179;  56.0 

.00546 

101  .4 

.0201 

49-8 

.00613 

90.2 

6.00 

19.6^ 

64.6 

.00473 

116.9 

.0176  56.8 

•00536 

102.9 

.0197 

50-7 

.00600 

91.8 

6.50 

21.33 

•  0153 

65.2 

.00466 

118.1 

•0174'  57-6 

-00530 

04-3 

.0194 

51-6 

.00591 

93-5 

7.00 

22.96 

.0152 

65.8 

.00463 

119.2 

.0172:  58.3 

.00524 

05.6 

.0191 

52-3 

.00582 

94-7 

7-50 

23.61 

.0151 

66.4 

.00460 

120.3 

.0170  58.9 

.00518 

06.7 

.0189 

53-° 

.00576 

96.0 

8.00 

26.25 

.0150 

66.9 

.00457 

121.  2 

.0168!  59.5 

.00512 

07.8 

.0186 

53-7 

.00567 

97-3 

8.50 

27.89 

.0149 

67-4 

.00454 

122.  1 

.o'66:  60.  i 

.00506 

08.9 

.0184 

54-3 

.00561 

98.4 

9  oo 

29-53 

.0148 

67.8 

.00451 

122.8 

.0165  60.7 

.0050- 

09.9]  .0182 

54-9 

.  00555  j    99.4 

9-50 

.0147 

68.2 

.00448 

123-5 

.0163:  61.2 

.00497 

10.8  .0180 

55.6 

.00549 

00.7 

lo.oo 

32.81 

.0146 

68.5 

.00445 

124.0 

.0162!  61.6 

.00494 

ii.  6  .oi7C 

56.0 

.00545 

01.4 

1  1  .00 

36.09 

.0144 

69.2 

.00438 

125.3 

.0160  62.5 

.00488 

13.2  .0176 

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MANNING'S    VALUES    OF  c    IN    THE    FORMULA    v 
UNIT   BEING    A   METRE    OR   A    FOOT. 


Value  of  m. 

Very  Smooth 
Surface. 

Smooth  Surface. 

Surface  not  very 
Smooth. 

Rough  Surface. 

Metre. 

Foot. 

Metre. 

Foot. 

Metre.  ' 

Foot. 

Metre. 

Foot. 

Metre. 

Foot. 

•05 

.16 

61 

no 

47 

85 

36 

65 

30 

54 

.  IO 

•33 

68 

123 

52 

94 

40 

72 

34 

62 

.20 

.66 

76 

137 

59 

107 

45 

81 

3S 

69 

•30 

.98 

82 

I48 

63 

114 

48 

87 

41 

74 

•50 

1.64 

89 

161 

69.         125 

52 

94 

45 

81 

I  .OO 

3-28 

TOO 

181 

77          "39 

59 

107 

50 

9i 

2.OO 

6.56 

112 

203 

86 

156 

66 

119 

56 

IOI 

3.00 

9.84 

120 

217 

92 

167 

71 

128 

60 

109 

5-oo 

16.40 

131 

237 

IOI 

183 

77 

139 

65 

118 

15.00 

49.20 

157 

284 

121 

219 

92 

167 

79 

143 

Value  of  m. 

Surface  in  Earth. 

Surface  in 
Gravelly  Soil. 

Irregular  Surface. 

Very  Irregular 
Surface. 

Metre. 

Foot. 

Metre. 

Foot. 

Metre. 

Foot. 

Metre. 

Foot. 

Metre. 

Foot. 

•05 

.16 

24 

41 

20 

36 

17 

31 

15 

27 

.IO 

-33 

27 

49 

23 

42 

19 

34 

17 

31 

.20 

.66 

31 

56 

25 

45 

22 

40 

19 

34 

•30 

.98 

33 

60 

27 

49 

23 

42 

2O 

36 

•50 

1.64 

36 

65 

30 

54 

25 

45 

22 

40 

I.OO 

3-28 

40 

72 

33 

60 

29 

53 

25 

45 

2.OO 

6.56 

45 

81 

37 

67 

32 

58 

28 

5i 

3-oo 

9.84 

48 

87 

40 

72 

34 

62 

30 

54 

5.00 

16.40 

52 

94 

44 

79 

37 

67 

33 

60 

15.00 

49.20 

63 

114 

52 

94 

45 

81 

63 

114 

328  EXAMPLES. 


EXAMPLES. 

1.  What  fall  must  be  given  to  a  canal  2600  ft.  fong,  7  ft.  wide  at  the 
top,  3  ft.  wide  at  the  bottom,  i|  ft.  deep,  and  conveying  40  cu.   ft.  of 
water  per  second  ?     (/  =  *V)  Ans.  i  in  135. 

2.  Determine  the  fall  of  a  canal    1500  ft.  long,  of  2   ft.   lower,  8  ft. 
upper  breadth,  and  4  ft.  deep,  which  is  to  convey  70  cu.  ft.  of  water  per 
second?     (/=  .008.)  Arts,  i  in  1088.4. 

3.  For  a  distance  of  300  ft.  a  brook' with  a  mean  water  perimeter  of 
40  ft.  has  a  fall  of  9.6  ins.  ;  the  area  of  the  upper  transverse  profile  is 
70  sq.  ft.,  that  of  the  lower  60  sq.  ft.      Find  the  discharge.     (/=  .008.) 

Ans.  352.12  cu.  ft.  per  sec. 

4.  In  a  horizontal  trench  5  ft.  broad  and  800  ft.  long  it  is  desired  to 
carry  off  20  cu.  ft.  discharge  and  to  let  it  flow  in  at  a  depth  of  2  ft.  ; 
what  must  be  the  depth  at  the  end  of  the  canal  ?     (/  —  .008.) 

Ans.  1.36  ft. 

5.  Water  flows  along  an  open  channel  12  ft.  wide  and  4  ft.  deep,  at 
the  rate  of  2  ft.  per  second.     What  is  the  fall?     A  dam  12  ft.  by  3  ft. 
high  is  formed  across  the  channel ;  how  high  will  the  water  rise  over  the 
crest  of  the  dam  ?  Ans.  i  in  480,  /  being  .08  ;  1.899  ft- 

6.  A  stream  is  rectangular  in  section,  12  ft.  wide,  4  ft.  deep,  and  falls 
i  in  loo.     Determine  the  discharge  (i)  with  an  air-perimeter;  (2)  without 
air-perimeter.     (/  =  .008.)  Ans.  (i)  646  cu.  ft.  per  sec. 

(2)  665.088  cu.  ft.  per  sec. 

7.  A  canal  20  ft.  wide  at  the  bottom  and  having  side  slopes  of  i|  to 
i  has  8  ft.  of  water  in  it ;  find  the  hydraulic  mean  depth. 

Ans.  5.163  ft. 

8.  The  water  in  a  semicircular   channel  of  10  ft.  radius  when  full 
flows  with  a  velocity  of  2  ft.  per  second;  the  fall  is  i  in  400.     Find  the 
coefficient  of  friction.  Ans.  .2. 

9.  Calculate  the  flow  per  minute  across  a  given  section  of  a  rectan- 
gular canal  20  ft.  deep,  45  ft.  wide,  the  slope  of  the  bed  being  22  ins.  per 
mile  and  the  coefficient  of  friction  per  square  foot  =  .008. 

Ans.  292.856  cu.  ft. 

10.  Why  does  the  water  of  a  river  rise  on  the  formation  of  the  ice  ? 

11.  Find  the  depth  and  width  of  a  rectangular  stream  of  900  sq.  ft. 
sectional  area,  so  that  the  flow  might  be  a  maximum  ;  also  find  the  flow, 
f  being  .008  and  the  slope  22  ins.  per  mile. 

Ans.  21.21  ft.  ;  42.42  ft. ;  4885  cu.  ft.  per  sec. 

12.  The  section  of  an  aqueduct  is  a  trapezium  with  a  bottom  width 
of  6.56  ft.,  a  top  width  of  7. 546  .ft.,  and  a  depth  of  7.874  ft.,  the  slope  is 
6  per  1000,  and  the  faces  of  the  aqueduct  are  of  brickwork.     Determine 


EXAMPLES.  329 

the  discharge  in  cubic  feet  per  second  when  the  depth  of  the  water  is 
4.92  ft.,  using  the  coefficient  given  by  (a)  Bazin  ;  (b)  Kutter ;  (c)  Man- 
ntng.  Ans.  (a)  471.276  ;  (b)  494.5484  ;  (c)  487.6973. 

13.  An  aqueduct  of  rectangular  section  is  to  convey  9504  (Imp.)  gal- 
lons of  water  per  hour  at  the  maximum  velocity  of  flow.     Assuming  as 
a  first  approximation  that  b  =  .000114,  and  tnat  the  slope  is  .33  per  1000, 
find  the  proper  width  and  slope.     Also  find  the  corresponding  velocity 
of  flow.  Ans.  i. 01  ft.  and  3  in  10,000;  .828  ft.  per  sec. 

14.  What  head   is  required  to  give  a  velocity  4  ft.  per  second  in  a 
semicircular  channel  of  3  ft.  diameter  and  5000  ft.  long,/ being  .0064? 

Ans.  10^  ft. 

i  5.  The  section  of  a  length'  of  La  Roche  Canal  in  rock  has  a  bottom 
width  of  .7  m.,  one  vertical  face  and  the  other  face  inclined  to  the  hori- 
zen  at  an  angle  tan"1  2.  The  mean  velocity  of  flow,  when  the  water 
runs  .5  m.  deep,  is  .514  m.  per  second.  Find  the  slope,  a  suitable  value 
for  the  coefficient  b  or  c  being  selected  from  the  Tables.  Ans.  .002. 

16.  A  section    of  the    La  Roche  Canal  in  earthwork  has    its  sides 
sloped  at  45°  and  has  a  bottom  width  of  .3  m.     When  the  depth  of  the 
water  is  0.5   m.   the  discharge  is  at  the  rate  of  205    litres  per  second. 
Determine  the  slope,  a  suitable  value  for  the  coefficient  b  being  selected 
from  the  Tables.     Also  show  that,  according  to   Bazin's  formula,  the 
maximum    surface   and   the  bottom  filament  velocities  are  .816  m.  and 
,49  m.,  respectively.  Ans.  Slope  =  .002. 

17.  Water  flows  along  a  symmetrical  channel,  20  ft.  wide  at  top  and 
&  ft.  wide  at  bottom  ;  the  friction  at  the  sides  varies  as  the  square  of  the 
velocity,  and  is   i   Ib.  per  square  foot  for  a  velocity  of  16  ft.  per  second. 
Find  the  proper  slope  so  that  the  water  may  flow  at  the  rate  of  2  ft.  per 
second  when  its  depth  is  6  ft.  Ans.  i  in  3445. 

1 8.  Calculate  the  flow  across  the  vertical  section  of  a  stream  4  ft. 
deep,  18  ft.  wide  at  top,  6  ft.  wide  at  bottom,  the  slope  of  the  surface 
being  18  in.  per  mile,     (f  =  .008.)  Ans.  110.9376  cu.  ft.  per  sec. 

19.  The  waterway  in  a  channel  of  a  regular  trapezoidal  section,  has  a 
sectional  area  of  TOO  sq.  ft.     If  the  banks  slope  at  40°  to  the  horizontal, 
what  will  be  the  best  dimensions  for  the  section  ? 

Ans.  Bottom  width  =  5.25  ft.  ;  depth  of  water  ='  7.22  ft. 

20.  The  sides  of  an  open  channel  of  given  inclination  slope  at  45° 
and  the  bottom  width   is   20  ft.     Find  the  depth  of  water  which  will 
make  the  velocity  of  flow  across  a  vertical  section  a  maximum. 

Ans.  6.73  ft. 

21.  The  banks  of  a  channel  slope  at  45°;  the  flow  across  a  transverse 
section  is  to  be  at  the  rate  of  loo'cubic  feet  at  a  maximum  velocity  of  5 
ft.  per  second.     Determine  the  dimensions  of  the  transverse  profile. 

Ans.  11.05  ft.  wide  at  bottom  ;  2.28  ft.  deep. 

22.  What   dimensions   must  be  given  to  the  transverse  profile  of  a 


33°  EXAMPLES. 

canal  whose  banks  slope  at  40°,  and  which  has  to  conduct  away  75  cubic 
feet  with  a  mean  velocity  of  3  ft.  per  second? 

Ans.   Depth  —  3.6  ft.;  width  at  bottom  =  2.62  ft. 

23.  The  section  of  a  canal  is  a  regular  trapezoid  ;    its  slope  is  i  in 
500;  its  width  at  the  bottom  is  8  ft.;  the  sides  are  inclined  at  30°  to  the 
vertical.     On  one  occasion   when  the  water  was  4  ft.  deep  a  wind  was 
blowing  up  the  canal,  causing  an  air-resistance  for  each  unit  of  free  sur- 
face equal  to  one  fifth  of  that  for  like  units  at  the  bottom  and  sides, 
where  the  coefficient  of  friction  may  be  taken  to  be  .08.     Determine  the 
discharge.  Ans.  75.34  cu.  ft.  per  sec. 

24.  A  canal  is  20  ft.  wide  at  the  bottom,  its  side  slopes  are  i|  to  i,  its 
longitudinal  slope  is  i  in  360;  calculate  H.M.D.  and  the  flow  per  minute 
across  any  given  vertical  section  when  there  is  a  depth  of  8  ft.  of  water 
in  the  canal.     (Coeff.  of  friction  =  .008.) 

If  a  weir  2  ft.  high  were  built  across  the  canal,  what  would  be  the 
increase  in  the  depth  of  the  water  ? 

Ans.  5.24  ft.;  2762.7776  cu.  ft.  per  sec.;  2.79  ft. 

25.  In  the  Ourcq  canal  the  earthen  banks  slope  at  cot~J    i|,  and  the 
bottom  width  is  3.5.    Find  the  depth  of  the  water  when  the  discharge  is 
3000  litres  per  second,  the  slope  of  the  canal  being  .1236  per  1000.    Also 
find  the  mean  velocity.  Ans.  1.5  m.  to  1.4  m.;  .4  m.  per  sec. 

26.  The  banks  of  a  canal  slope  at  45°,  the  section  being  a  trapezium. 
The  discharge  is  to  be  1200  litres  per  second  at  the  rate  of  .5  m.  per 
second.     Find  the  best  bottom  width  and  depth  and  also  the  slope. 

Ans.  .94  m.;  1.14  m.;  .0004  according  to  Bazin  and  .0003  accord- 
ing to  Manning,  the  mean  being  .00035. 

27.  In  the  transverse  section  ABCD  of  an  open  channel  with  a  ver- 
tical slope  of  i  in  300,  the  bottom  width  is  20  ft.,  the  angle  ABC  =  .90° 
and  the  angle  BCD  =  45°.     Find  the  height  to  which  the  water  will 
rise  so  that  the  velocity  of  flow  may  be  a  maximum  ;  also  find  the  dis- 
charge across  the  section,  f  being  .008. 

Ans.  11.715  ft.;  1584  cu.  ft.  per  second. 

28.  The  sewers  in  Vancouver  are  square  in  section  and  are  laid  with 
one  diagonal  vertical.     To  what  height  should  the  water  rise  so  that  (a) 
the  velocity  of  flow  may  be  a  maximum;  (b~)  the  discharge  may  be  a 
maximum?     (A  side  of  the  square  —  12  in.) 

Ans.  (a)  .292  ft.  above  horizontal  diameter. 
(<*)•  5797  ft.     " 

29.  The  section  of  a  channel  is  a  rhombus  with  a  diagonal  vertical. 
How  high  must  the  water  rise  in  the  channel  (a)  to  give  a  maximum  of 
flow,  and  (ff)  to  give  a  maximum  discharge? 

Ans.  .If  D  is  the  length  of  the  horizontal  diameter,  and  if  0  is 
the  inclination  of  a  side  to  the  vertical,  the  water  must  rise  above 
the  horizontal  diameter  to  the  height  D  cot  0  x  .207  in  (a}  and 
to  the  height  D  cot  6  x  .4099  in  (ft). 

30.  An  aqueduct  has  a  given  slope  and  a  square  section  with  a  diag- 


EXAMPLES.  33  r 

onal  vertical.  Show  that  the  discharge  at  maximum  velocity,  the  dis- 
charge when  running  full  and  the  maximum  discharge  are  in  the  ratios 
of  i  to  1.  115  to  1.140,  and  that  the  corresponding  mean  depths  are 
.293^,  .25^,  and  .27^,  a  being  a  side  of  the  square. 

31.  An  aqueduct,  with  a  section  in  the  form  of  an  isosceles  right- 
angled  triangle  of  height  h,  is  laid  with  its  base  horizontal.  Compare 
the  quantities  of  water  conveyed  (a)  when  running  full  ;  (&)  when  the 
velocity  is  a  maximum;  (<r)  when  the  quantity  conveyed  is  a  maximum, 
and  find  the  corresponding  mean  depths. 

Ans.   Quantities,       (a}  <-***;         0)  £;         (,) 

}  '  } 


. 
2.1973  2.0906  2.0484 

Mean  depths,  (a)  .207/2;          (b)  .2288/1  ;         (c)  .218/2. 

32.  A  length  of  a  circular  aqueduct  of  waterway  A  and  mean  depth 
m  has  to  be  replaced  by  a  length  of  an  equivalent  rectangular  aqueduct. 
If  the  depth  of  the  water  is/  and  the  width  of  the  rectangular  section  x, 
show  that 


the  value  of  —  being  b'  for  the  pipe  and£  for  the  rectangular  aqueduct. 

Note.  —  In  first  approximations  take  b  —  b'  . 

33.  Taking  the    coefficient    b    for    a    given    open   channel   to    be 

.00010058  and  the  corresponding  coefficient  (  =  —  )  for  pipe-flow  to  be 

.00012485,  show  that,  approximately,  if  the  volume  of  flow  under  the  same 
head  is  the  same  both  for  the  channel  and  the  pipe, 

d*P  =  8  A\ 

A  being  the  sectional  area  of  the  waterway  in  the  channel,  P  the  wetted 
perimeter  and  d  the  diameter  of  the  pipe. 

34.  Using  the  same  coefficients  as  in  the  preceding  example,  show 
that  the  loss  of  head  per  unit  of  length  in  a  pipe  is  nearly  88  per  cent 
greater  than  the  loss  in  an  open  semicircular  channel  of  an  equal  water- 
way and  giving  the  same  discharge. 

(Note.—  Since  the  whole  of  a  pipe-surface  develops  resistance  to  flow,  it 
is  evident  a  priori  that  the  loss  of  head  per  unit  of  length  imist  be  much 
greater  than  in  the  case  of  the  open  channel?) 

35.  The  Dhuis  aqueduct,  which  supplies  Pau  with  water,  has  a  slope 
of  i  in  10,000.    Its  section  is  egg-shaped,  the  lowest  portion  being  a  semi- 
circle of  .7  m.  radius.     The  aqueduct  conveys,  normally,  200  litres  per 
second.    Find  the  angle  subtended  at  the  centre  of  the  semicircle  by  the 
water-line,  and  hence  find  the  sectional  area  of  the  waterway,  its  depth 
and  the  velocity  of  flow.     Ans.  154°;  .55  sq.  m.;  .54  m.;  36  m.  per  sec. 


33 2  EXAMPLES. 

36.  Deduce   the  flow  formula  for  a  circular  aqueduct  of  radius  rt 
when  the  wetted  perimeter  subtends  an  angle  of  240°  at  the  centre. 

Ans.  r*i  =  .26il>Q*. 

37.  A  circular  aqueduct  of   6.56  ft.  diam.  conveys  49.44  cu.  ft,  of 
water  per  sec.     The  slope  is  i  in   10,000.     Find  (a)  the  angle  subtended 
at  the  centre  by  the  water-line ;  (b)  the  clear  head  above  the  water  sur- 
face ;  (c)  the  velocity  of  flow. 

Ans.  (a)  240°  30'  ;  ( b]  1.63  ft.;  (c)  1.815  ft-  Per  sec- 

38.  The  Avre  circular  aqueduct  conveys  2.05  cm.  per  second,  and  in 
one  length  the  slope  is  4  in  10,000.     Its  water-line  subtends  120°  at  the 
centre.     Find  the  radius,  taking  b  —  .0002  as  a  first  approximation. 

The  surface  has  a  very  smooth  coat  of  cement  .02  in.  thick;  de- 
termine the  actual  waterway,  the  wetted  perimeter,  the  mean  depth, 
the  velocity  of  flow,  and  the  clear  height  above  the  water-line. 

Ans.  Radius  =  .88  m. ;  1813  sq.  m. ;  3.549  m. ,  .51  m. ;  1.13  m. 
per  sec. ;  .445  m. 

39.  The  Potomac  aqueduct,  which  is  faced  with  brick,  has  a  diameter 
of  9.0225  ft.  and  a  slope,  of  .143  in  10,000.     The  water-line  subtends  an 
angle  of  240°  at  the  centre.     Taking  b  =  .0000609,  determine  quantity  of 
water  conveyed  in  gallons  per  day.  Ans.  69,997,071  Imp.  gallons. 

84,019,066  U.  S.  gallons. 

40.  Taking  b  =  .0000609,  find  the  angle  subtended  at  the  centre  by 
the  water-line  and  also  find  the  free  height  above  the  water-surface   in 
the    Vanne   aqueduct  when   conveying  49.442    cu.   ft.    per   second,    the 
diameter  of  the  aqueduct  being  6.562  ft.,  and  the  slope  i  in  10,000. 

Ans.   240°  30'. 

41.  Show  that  the  quantities  of  water  conveyed  by  a  circular  aque- 
duct of  radius  r,  when  the  water-line   subtends  an  angle  of  240°  at  the 
centre,  when   the   velocity  of  flow   is  greatest,  when  running  full,  and 
when  the  quantity  conveyed  is  a  maximum,  are  in  the  ratios  of  i  to  1.086 
to  1.131    to   1.188,  and  find  the  angles  subtended  at  the  centre  by  the 
water-lines  in  the  three  last  cases.     Also  determine  the  mean  hydraulic 
depths.  Ans.  Angles,  257°  27'  ;  360° ;  308°. 

Mean  depths,  .603^ ;  .6o8r;  .5^;  .573^. 

42.  For  a  small  tachometer  the  velocities  are  .163,  .205,  .298,  .366, 
.61  metre  ;    the   number   of  revolutions  per    second   are   .6,  .835,    1.467, 
1.805,  3-l42-     Find  the  constants  corresponding  to  the  wheel. 

Ans.  .169;  .061. 

43.  Assuming  (i)  that  a  river  flows  over  a  bed  of  uniform  resistance 
to  source;    (2)  that  to  maintain  stability  the  Velocity  is  constant  from 
source  to  mouth;  (3)  that   the  river  sections  at  all  points  are  similar ; 
(4)  that  the  discharge  increases  uniformly  in  consequence  of  the  supply 
from  affluents — determine  the  longitudinal  section  of  such  a  river. 

Ans.  A  parabola. 


EXAMPLES.  333 

44.  In  an  aqueduct  with  a  slope  of  i  in   10,000,  the  depth  of  water 
corresponding  to  a  condition  of  uniform  steady  motion  is  1.77  ft.     At 
a  certain  point  the  depth  is  increased  to  4.43   ft.  by  a  weir  3.77  ft.  in 
height.     Find    the   distance   to  which   the  "rise"   extends   along   the 
aqueduct.  Ans.  50,038  ft. 

45.  The  channel  of  a  river  328  ft.  wide  is  narrowed  by  the  abutments 
of  a  bridge  to   a  width  of  42.65  ft.     The  depth  of  the  water  under  the 
bridge  is  12.63  &•>  an<^  tne  quantity  of  flow  per  hour  is  2,406.250  gallons. 
Find  the  height  of  swell.  Ans.  .104  ft. 

46.  In  a  broad  channel  of  approximately  rectangular  section   there  is 
a  small   change  of   n%  in  the   depth.      Show    that -the   corresponding 
changes  in  the  velocity  of  flow  and  in  the  discharge  are  \n%  and   \\n% 
respectively.     Also,   if  the  banks   slope  at.  an  angle  6,  show  that  the 

nhvl  b  i       \         ,    nhQl  ^b  i      \ 

changes   become   -         --  —  -T—      -       and    —      — -  —  — — : — -       respec- 
100  \zA       P  sin  6/  100  \2A        P  sin  6/ 

tively,  /i,  b,  A,  P,  v,  and  Q  being  the  initial  depth,  breadth,  area  of  water- 
way, wetted  perimeter,  velocity  of  flow,  and  discharge,  respectively. 


CHAPTER    IV. 


RAMS,    PRESSES,   ACCUMULATORS,   WATER-PRESSURE 

ENGINES. 

I.  Hydraulic  Rams. — By  means  of  the  hydraulic  ram  a 
quantity  of  water  falling  through  a  vertical  distance  hv  is  made 
to  force  a  smaller  weight  of  water  to  a  higher  level. 

The  water  is  brought  from  a  reservoir  through  a  supply- 
pipe  5.  At  the  end  of  this  pipe  there  is  a  valve  opening  into 
an  air-chamber  C ,  which  is  connected  with  a  discharge-pipe  D. 
At  E  there  is  a  weighted  check-  or  clack-valve  opening 
inwards,  and  the  length  of  its  stem  (or  the  stroke)  is  regulated 
by  means  of  a  nut  or  cottar.  When  the  waste-valve  at  E  is 
open  the  water  begins  to  escape  with  a  velocity  due  to  the 
head  //L  and  suddenly  closes  the  valve.  The  momentum  of 

the  water  in  the  pipe  opens 
the  valve  at  B,  and  a  por- 
tion of  the  water  is  dis- 
charged into  the  air- vessel. 
From  this  vessel  it  passes 
into  the  discharge-pipe  in 
consequence  of  the  reac- 
tion of  the  compressed  air. 
At  the  end  of  a  very  short 
interval  of  time  the  mo- 


FIG. 179. 


mentum  of  the  water  has 
been  destroyed,  the  valve 
opening  into  the  chamber  C  closes,  the  waste-valve  again 
opens,  and  the  action  commences  as  before.  It  is  found  that 

334 


HYDRAULIC  PRESS.  335 

the  efficiency  of  the  ram  is  increased  by  introducing  the  small 
air-vessel  F.  The  wave-motion  started  up  in  the  supply-pipe 
by  the  opening  and  closing  of  the  valve  opening  into  the 
chamber  C,  has  been  utilized  in  driving  a  piston  so  as  to  pump 
up  water  from  some  independent  source. 

Let  v  be  the  velocity  of  flow  in  the  supply-pipe  at  the 
moment  when  the  valve  at  E  is  closed. 

Let  Wl  be  the  weight  of  the  mass  of  water  in  motion. 

W  v  3 
Then  -       L   is  the  energy  of  the  mass,  and  this  energy  is 

expended  in  opening  the  valve  at  B,  forcing  the  water  into  the 
air-chamber,  compressing  the  air,  and  finally  causing  the 
elevation  of  a  weight  W2  of  the  water  through  a  vertical  dis- 
tance h' . 

Let  hj  be  the  head  consumed  in  frictional  and  other 
hydraulic  resistances. 

Then 

W  vz 
W2(hf  +  /y)  =  the  actual  work  done  =  — -  — . 

This  equation  shows  that,  however  great  //  may  be,  Wa 
has  a  definite  and  positive  value,  and  therefore  water  may  be 
raised  to  any  required  height  by  the  hydraulic  ram. 

WJt' 

The  efficiency  of  the  machine  =       * .  ,  and  may  be  as  much 

w\ll\ 

as  66  per  cent  if  the  machine  is  well  made.  According  to 
d'Aubuisson, 


2.  Hydraulic  Press. — The  hydraulic  press  is  a  machine 
by  means  of  which  great  pressures  can  be  exerted  and  heavy 
weights  lifted,  the  energy  being  transmitted  through  water. 
It  consists  essentially  of  a  strong  cast-iron  or  cast-steel  cham- 
ber or  cylinder  containing  a  plunger  or  ram  which  is  acted 


336 


HYDRAULIC  PRESS. 


upon  by  water  pumped  through  piping  into  the  chamber  by  a 
single-acting  force-pump,  which  may  be  either  worked  by 
hand  or  by  power. 

The  action  of  the  press 
depends  on  the  principle 
that  fluids  press  equally  in 
all  directions  and  thus  the 
pressure  per  square  inch 
on  the  ram  is  equal  to  the 
pressure  per  square  inch  on 
the  pump-plunger.  Origi- 
nally discovered  by  Pascal, 
FIG.  180.  the  press  was  first  made  of 

practical  utility  by  Bramah,  who  made  the  moving  parts  water- 
tight by  the  introduction  of  cup-leather  packing. 

The  ram  is  packed  with  a  leather  collar  of  f|  form  which 
is  fitted  into  a  recess  turned  out  in  the  neck  of  the  cylinder  and 
is  kept  in  place  by  the  cylinder-cover  gland. 
According  to  experiments  made  by  Hick, 
the  friction  at  the  collar  increases  directly 
with  the  diameter  of  the  ram  and  with  the  FlG- 


pressure,  but  is  independent  of  the  depth  of  the  collar, 
law  of  friction  is  expressed  by  the  following  formula: 


Hick's 


the  total  frictional  resistance  =  .0314^  or  .0471^, 

according  as  the  leather  is  in  good  condition  and  well  lubri- 
cated or  is  new  and  badly  lubricated. 

The  friction  is  about  I  per  cent  of  the  pressure  for  a  4-in.  ram. 

At  low  pressures  hemp  packing  is  invariably  used,  and 
sometimes  also  for  pressures  as  great  as  2000  Ibs.  per  sq.  in., 
but,  generally  speaking,  it  is  rarely  used  for  pressures  exceed- 
ing about  700  Ibs.  per  sq.  in.  The  ram  is  driven  forwards  by 
the  pressure  of  the  water  through  the  tight  collar,  and  is  capable 
of  lifting  a  weight  or  exerting  a  pressure  which  is  limited  in 


HYDRAULIC  PRESS.  337 

magnitude  only  by  the  strength  of  the  chamber  and  connec- 
tiogs  and  by  the  capacity  of  the  pump. 

Let  L  be  the  stroke  of  the  ram. 

Let  W  be  the  weight  on  the  ram,  including  the  weight  of 
the  ram. 


Then  the  work  done  = 

Let  Q  be  .the  axial  force  on  the  plunger  produced  by  a  force 
P  on  the  pump-lever  at  a  distance  p  from  the  fulcrum. 

Let  q  be  the  distance  between  the  fulcrum  and  the  axis  of 
the  plunger.  Then,  disregarding  fluid  friction,  the  friction  at 
the  fulcrum,  and  the  leather  or  '  '  packing  '  '  friction, 

Pp  =  07. 

irD*          *D>    W       D'  Pp 

But  ^—^—-W- 


If  rl ,  r0  are  the  internal  and  external  radii  of  a  press,  and 
if  p^ ,  pQJ  arid  ,/are  the  intensities  of  pressure  at  the  internal 
and  external  surfaces  and  the  intensity  of  stress  at  the  radius 
r,  then 

f_Pf*      *"* 


so'  o       jf\'  i ,    SQ       r\      '  o'  i 

ro         ri  r~        ;V        r\ 

(See  Appendix,  "Th.  of  Structures,"  Bovey.) 

Hydraulic  presses  of  different  designs,  but  which  are  all 
more  or  less  modifications  of  the  Bramah,  are  employed  for  a 
variety  of  pressing  and  lifting  operations.  For  example,  they 
are  used  in  making  lead  pipes,  in  expressing  oil  from  seeds,  in 
baling  cotton,  in  pressing  yarn,  in  packing  hay,  etc.,  while 
the  modern  systems  of  punching,  riveting,  stamping,  forging. 


338 


HYDRAULIC  JACK. 


shearing,  welding,  and  bending  depend  upon  the  peculiar 
advantages  of  hydraulic  power  for  such  purposes,*  Hydraulic 
presses  for  forging  have  largely  superseded  the  steam-hammer. 


FIG.  182. 
Hydraulic  Press. 


FIG.  183. 
Portable  Riveter. 


FIG.  184. 
Baling-press. 


and  it  is  now  common  to  find  presses  with  capacities  ranging 
from  4000  to  10,000  tons,  the  working  intensity  of  pressure 
being  as  great  as  3  tons  per  sq.  in. 

The  hydraulic  jack,  Fig.  185,  is  a  portable  machine  for 
raising  heavy  weights  through  short  distances.  It  is  a  com- 
pact combination  of  a  force-pump  and  a  press.  The  ram  Q 
fits  the  press  5"  and  is  made  water-tight  by  the  cup-leather  D. 
The  pump  is  worked  by  the  up-and-down  movement  of  a 
lever  which  presses  upon  a  cam  or  is  connected  with  other 
suitable  gearing  and  communicates  motion  to  the  pump-plunger 
.R.  The  water  in  the  chamber  is  thus  forced  through  a  valve 
into  the  hydraulic  cylinder,  developing  a  pressure  which  causes 
the  ram  to  rise  and  to  lift  the  load  resting  on  the  head  H. 
*  In  America  compressed  air  is  largely  used  for  punching,  riveting,  etc. 


ACCUMULATOR. 


339 


As  the  pump-plunger  rises  a  partial  vacuum  is  produced 
in  the  pump-chamber,  and  the  pressure  in 
the  reservoir  B  overcomes  the  resistance  of 
the  spring  on  the  inlet-valve  and  opens  a 
passage  for  the  water  into  the  pump-cham- 
ber. To  lower  the  jack,  a  relief- valve  is 
unscrewed,  and  the  water  returns  to  the 
reservoir  B  while  the  ram  falls.  The  ram 


FIG.  185. 


FIG.  186. 


may  be  prevented  from  turning  round  by  .means  of  a  steel  set- 
pin  screwed  on  the  side  of  the  press  and  fitting  a  vertical  slot 
in  the  ram. 

The  construction  and  action  of  the  punching-bear,  Fig. 
1 86,  are  essentially  the  same  as  in  the  hydraulic  jack.  By 
actuating  the  lever  Z,  the  water  passes  into  the  hydraulic 
cylinder  C  and  by  its  action  forces  the  punch  P  down.  The 
punch  is  raised  by  first  opening  a  relief-valve  and  then  lower- 
ing the  leVer  M,  which  causes  the  cam  to  raise  the  hydraulic 
ram,  and  the  water  from  the  hydraulic  cylinder  flows  back 
into  the  reservoir.  The  relief-valve  is  now  closed  and  the 
punching  operation  may  be  again  repeated. 

3.  Accumulator. — Low  pressures  of  170  Ibs.  (=  392  ft.) 
to  250  Ibs.  (=  576  ft.)  per  sq.  in.  can  sometimes  be  obtained 
from  a  natural  supply  or  from  a  reservoir,  but  the  higher 


34° 


ACCUMULATOR. 


pressures  of  700  Ibs.  (=  1612  ft.)  to  1000  Ibs.  (=  2304  ft.) 
per  sq.  in.  and  upwards,  which  are  almost  exclusively  adapted 
to  the  working  of  intermittent  machines,  must  be  artificially 
produced  by  means  of  pumping-engines.  In  a  direct  supply 
the  capacity  of  these  engines  must  be  sufficient  to  meet  the 
maximum  demand  at  any  moment,  but  the  fluctuation  in  the 
demand  upon  the  mains  for  cranes,  capstans,  elevators,  etc., 
was  soon  found  to  be  so  great  as  to  render  imperative  some 
method  of  storing  energy.  This  has  been  effected  by  the 
introduction  of  the  accumulator,  which,  in  its  simplest  form, 
consists  of  an  annular  cylinder  (Fig.  187)  partially  or  wholly 
filled  with  scrap,  slag,  or  other  heavy  material,  or  of  a  series 
of  trays  (Fig.  188)  loaded  with  pig  iron  or  lead,  supported  by 
a  cross-head  on  the  top  of  a  ram  working  in  a  cylinder  with  a 


FIG.  187. 


FIG.  188. 


stuffing-box  and  gland  at  the  upper  end.  The  pressure -water 
is  admitted  by  a  branch  pipe  at  the  lower  end  and  raises  the 
ram  together  with  the  weight  it  carries.  Thus,  if  JKtons  are 
.lifted  through  a  vertical  distance  s  and  if  the  water -pressure  on 


INTENSIFIED  34t 

the  ram  of  d  in.  diameter  is  p  Ibs.  per  sq.  in.,  the  total  store 
of  energy  in  foot-pounds 


=  224011  s  =  ---  sp. 
4 

When  the  accumulator  has  reached  the  highest  point  it 
actuates  a  lever  which  shuts  off  the  steam  so  that  the  engines, 
cease  to  work  and  the  accumulator  falls.  When  it  has  reached 
the  lowest  point  it  again  actuates  a  lever  which  opens  a  valve 
and  admits  steam.  The  engines  again  commence  to  work  and 
the  accumulator  rises. 

In  small  plants  the  accumulator  fully  provides  for  the 
storage  of  sufficient  energy  to  meet  the  momentary  fluctuations 
of  demand  for  the  power  necessary  to  work  machines  which 
are  intermittent  in  action,  and  without  the  accumulator  pump- 
ing-engines  of  greater  capacity  would  be  required.  In  large 
plants,  as  in  the  cities  of  London,  Manchester,  and  Glasgow, 
the  total  accumulator  storage  capacity  is  a  very  small  fraction 
of  the  total  supply,  and  at  the  times  when  the  demand  is 
heavy  the  accumulators  are  usually  almost  stationary.  In  such 
cases  they  may  be  considered  rather  as  regulators  of  pressure. 
They  are  also  of  great  importance  in  automatically  facilitating 
the  control  of  the  plant,  and  act  as  buffers  in  preventing  break- 
age and  shocks.  If  lack  of  space  prevents  the  use  of  an  accu- 
mulator of  the  type  just  described,  an  intensifier,  Fig.  189, 
may  be  employed.  Water  at  a  pressure  of  p  Ibs.  per  sq.  in. 
is  admitted  from  the  water-mains  or  from  a  tank  at  a  suitable 
elevation  to  the  lower  side  of  a  piston  of  diameter  D  ins.,  work- 
ing in  an  hydraulic  cylinder.  The  piston-rod  of  diameter  d 
ins.  forms  the  ram  of  the  accumulator  B,  and  works  through  a 
water-tight  neck.  Thus  the  pressure  in  the  accumulator  in  Ibs. 
per  sq.  in. 

TlD* 


342 


DIFFERENTIAL   ACCUMULATOR. 


and  this  is  also  the  intensity  of  the  pressure  in  the  hydraulic 
mains  C. 

Tweddell's  differential  accumulator.  Fig.  190,  is  also 
designed  for  cases  in  which  space  is  of  importance.  A  heavy 
cylinder  A,  with  the  usual  glands  and  cup-leathers  at  the  top 


FIG.  189. 


FIG.  190. 


and  bottom,  is  loaded  with  a  number  of  lead  or  cast-iron 
weights  W,  fitted  into  each  other,  and  slides  upon  a  ram  />, 
fixed  at  the  upper  end  by  a  bracket  and  at  the  lower  by  a  step. 
A  brass  liner  is  shrunk  upon  the  lower  portion  of  the  ram*  so 
that  its  diameter  is  slightly  greater  than  that  of  the  upper 
portion.  A  hollow  passage  C  is  drilled  axially  along  the  ram 
and  connects  with  a  cross-passage  just  above  the  brass  liner. 
The  water  is  pumped  through  the  inlet-pipe  /,  fills  these 
passages  and  exerts  an  upward  pressure  over  an  effective  area 
equal  to  the  difference  between  the  areas  of  the  lower  and 
upper  portions  of  the  ram.  Thus  very  heavy  pressures,  up 
to  2OOO  Ibs.  per  sq.  in.,  or  more,  can  be  readily  obtained  with 
a  comparatively  small  weight.  But  the  volume  of  water  is 

*  The  ram,  however,  is  usually  solid  steel. 


DIFFERENTIAL   ACCUMULATOR.  343 

small,  and  any  large  demand  for  power  will  cause  the  loaded 
cylinder  to  fall  rapidly,  so  that  when  it  is  brought  to  rest  a 
considerable  increase  of  pressure  ;is  developed  which  is  of 
advantage  in  punching,  riveting,  etc.  The  uppermost  weight 
is  connected  by  means  of  a  chain  with  a  relief-valve  which 
enables  the  limiting  positions  of  the  cylinder  to  be  automati- 
cally regulated. 

Let  J^be  the  total  dead  weight  lifted. 

Let  F  be  the  friction  of  each  of  the  cup-leathers. 

Let  dv ,  d2  be  the  diameters  of  the  lower  and  upper  portions 
of  the  ram. 

With  the  cylinder  at  the  height  x  above  its  lowest  position, 
let  pv  be  the  intensity  of  pressure  in  the  inlet-pipe  /  when  the 
cylinder  is  rising,  and  p.2  the  intensity  when  it  is  falling.  Then 

W+  2F 

A  =  w*+  ~ — -» 

W-  2F 

A  =  «'•*+  — 

~W  -  O 

Hence  an  approximate  measure  of  the  variation  of  the 
intensity  of  pressure  is 

i6F 

Pi"p'=^dI«-d,y 

and  the  value  of  this  variation  is  ordinarily  from  about  I  per 
cent  of  the  pressure  for  a  i6-in.  ram  to  about  4  per  cent  for  a 
4-in.  ram. 

Experiment  has  shown  the  efficiency  of  an  accumulator  to 
be  as  high  as  98  per  cent,  I  per  cent  being  lost  in  charging 
and  i  per  cent  in  discharging.  Its  total  store  of  energy  is 
comparatively  small  and  it  cannot  maintain  a  supply  for  any 
length  of  time,  but  it  possesses  the  great  advantage  of  being- 
able  to  use  its  energy  at  a  high  rate  for  a  short  period. 


344 


HYDRAULIC  ENGINES. 


Fig.  191  represents  a  convenient 
form  of  accumulator  known  as  Brown's 
Steam  Accumulator.  A  ram  R  works 
in  the  hydraulic  chamber  //,  into 
which  water  is  forced  by  a  pair  of 
engines.  A  piston  P  is  attached  to 
the  upper  end  of  the  ram  and  works  in, 
a  cylinder  supplied  with  steam  direct 
from  the  boilers.  As  soon  as  pressure- 
water  is  supplied  to  hydraulic  ma- 
chinery the  ram  and  piston  fall,  opening 
the  steam-port,  so  that  steam  passes 
into  the  engine-cylinders.  The  pumps 
then  commence  to  work  and  force  in 
FIG.  191.  more  water  to  replace  that  which  is 

being  drawn  off.     This  accumulator  is  specially  for  use  on  ships. 
4.   Water-pressure    Engines.  —  In   these    engines   water 

under  pressure  is  admitted  into 

a  strong  chamber  or  cylinder, 

and     acts     upon     a     piston     or 

plunger   in   precisely  the   same 

manner  as  in   the  case   of   the 

steam-engine.      The  cylinder  is 

made  of  gun -metal  or   of   cast 

iron,  and  its  thickness  /,  which 

is   relatively   large    on    account 

of  the  wear,  may  be  calculated 

from  the  formula 

/ins.  =  .0024/V/+  1.25  ins., 

fa  being  the  pressure  in  atmospheres,  and  d  the  diameter  in 
inches. 

The  frictional  resistances  and  the  possibility  of  severe  shocks 
are  increased  by  rapid  motion  and  reversals  of  motion.  Hence 
the  velocity  of  flow  in  the  supply-pipe  should  not  exceed  10  ft. 


FIG.  192. 


HYDRAULIC   LIFTS. 


34S 


per  second,  and  preferably  should  be  limited  to  6  ft.  per 
second  (Art.  n,  p.  156),  while  the  plunger  should  have  a  long 
stroke.  In  practice  the  stroke  is  usually  from  2\  to  6  times 


FIG.  193. — Sectional  Elevation. 


FIG.  194. — Cross-section. 


FIG.  195.— Freight-hoist. 


FIG.  196.  —  Balanced-ram  Lift. 


the  diameter  of  the  cylinder,  and  the  mean  velocity  of  the 
plunger  is  about  I  ft.  per  second,  rarely  exceeding  80  ft.  per 
minute.  As  the  water  is  practically  incompressible,  its  free 
and  immediate  passage  should  be  insured  by  means  of  large 


346 


HYDRAULIC  LIFTS. 


and  wide-open  ports.  An  important  advantage  connected 
with  this  property  of  incompressibility  is  that  the  hydraulic 
resistances  may  be  indefinitely  increased  by  simply  closing  a 
valve.  Thus  no  brakes  are  required,  but  the  water  contains 
within  itself  its  own  brake,  and  an  absolute  control  is  provided 
which  secures  the  highest  degree  of  safety. 

The  water-pressure  engine  is  necessarily  a  slow-moving 
machine,  and  is  both  cumbrous  and  costly  unless  actuated  by 
pressures  of  great  intensity.  These  engines  are  advantageously 
employed  in  working  cranes,  hoists,  elevators,  capstans,  dock- 
gates,  presses,  and  other  machinery  in  which  the  action  is  of 
an  intermittent  character. 

The  hydraulic-ram  lift,  Fig.   197,  more  completely  utilizes 

than  any  other  the  properties  of 
incompressibility  and  direct  pres- 
sure, and,  owing  to  its  greater 
safety,  its  adoption  is  sometimes 
recommended  for  elevators  of  con- 
siderable height.  Under  a  full 
load  its  efficiency  may  be  as  great 
as  95  per  cent.  The  speed  of  a 
suspended  lift  is  rarely  less  than 
100  ft.  per  minute  and  often  ex- 
ceeds 500  or  600  ft.  per  minute. 
Between  such  limits  a  large  varia- 
tion in  the  efficiency  might  be 
expected,  and  although  the  effi- 
ciency under  a  full  load,  even 
when  the  ram-stroke  is  multiplied 
8  or  10  times,  may  be  75  or  8° 
per  cent,  it  may  also  fall  below 
40  per  cent  when  the  load  is  light. 
The  chief  loss  of  efficiency  is 
FlG<  IQ7<  due  to  the  fact  that  the  same 

quantity  of  pressure-water,   and  therefore  of  energy,  is  .used 


HYDRAULIC  ENGINE.  347 

\vhether  the  load  is  heavy  or.  light.  Various,  devices  have  been 
.adopted  to  remedy  this  evil  :  the  length  of  stroke  may  be 
automatically  proportioned,  as  in  the  Hastie  engine,  to  the 
work  to  be  done;  the  pressure-water  may  be  admitted  for  a 
part  of  the  stroke  only,  the  remainder  being  provided  by  the 
discharge-water;  cranes  and  elevators  are  often  provided  with 
a  large  cylinder  for  heavy  loads  and  a  small  cylinder  for  light 
loads,  and  for  the  same  purpose  a  single  cylinder  with  a  differ- 
ential piston  is  sometimes  used. 

Other  important  losses  of  efficiency  are  due  to  (a)  pipe 
friction;  (b)  elbows,  curves,  etc.,  and  abrupt  changes  of  sec- 
tion ;  (c)  the  friction  of  mechanism. 

Let  pm  be  the  mean  intensity  of  the  pressure  in  the 
cylinder. 

Let  j-  be  the  stroke. 

Let  i'm  be  the  mean  velocity  of  the  plunger. 

Then 

the  work  done  per  stroke  =  —  pms\ 

4 

the  quantity  of  motive  water  used  per  stroke 
•nd~  I   nd* 


according  as  the  engine  is  of  the  double-  or  single-acting  type. 

Analysis.  —  In  a  direct-acting  pressure-engine  let  A  be  the 

sectional  area  of  the  working  cylinder   (Fig. 


Let  a  be  the  sectional  area  of  the  supply- 
pipe. 

Let  A  =  na.  FlG'  I98' 

Let  Wbe  the  weight  of  the  water,  piston,  and  other  recip- 
rocating parts  in  the  working  cylinder. 

Let  /  be  the  length  of  the  supply-pipe. 

Let  f  be  the  acceleration  of  the  piston.      Then  nf  is  the 
acceleration  of  the  water  in  the  supply-pipe. 


348  HYDRAULIC  ENGINE. 

The  loree  required  to  accelerate  the  piston 

-  & 

and  the  corresponding  pressure  in  feet  of  water 

_^/ 
~  w  A  g  ' 

The  force  required  to  accelerate  the  water  in  the  supply 
pipe 

wal 


and  the  corresponding  pressure  in  feet  of  water 


A 

Similarly,  if  I'  is  the   length  of  the  discharge-pipe  and  —  . 

its  sectional  area,  the  pressure-head  due  to  the  inertia  of  the 
discharge-water 


Hence  the  total  pressure  in  feet  of  water  required  to  over- 
come inertia  in  the  supply-pipe  and  cylinder 


W 

The  quantity  —  A  -\-  nl  has  been  designated  the  length  of 
iv  A 

working  cylinder  equivalent  to  the  inertia  of  the  moving  parts. 
Let  the  engine  drive  a  crank  of  radius  r,  and  assume  that  the 
velocity  V  of  the  crank-pin  is  approximately  constant.  Then 
the  acceleration  of  the  plunger  when  it  is  at  a  distance  x  from 
its  central  position 


HYDRAULIC  ENGINE.  349 

and  the  pressure  due  to  inertia 


Let    i'    be    the    velocity   of  the    plunger   in    the   working 
cylinder. 

Let  u  be  the  velocity  of  the  water  in  the  supply-pipe. 

Let  //  be  the  vertical  distance  between  the  accumulator-ram 
and  the  motor. 

Let  /0  be  the  unit  pressure  at  the  accumulator-ram. 

Let  /  be  the  unit  pressure  in  the  working  cylinder. 

Then 

PQ        u2  p        i'2        (  losses    due     to    friction,    sudden 

w~^  2g  ~  w~^~  2g  -*~\      changes  of  section,  etc. 


Thus 

—  /2  — 

" 


W  2g 


-  //  +  losses. 


7/2  —   U2 

The  term  ----      +  losses  may  be  approximately  expressed 

7/2 

in   the  form  K      ,  K  being  the  coefficient  of  hydraulic  resist- 
ance.     Hence 


the  term  h  being  disregarded,  as  it.  is  usually  very  small  as 
compared  with  —  . 

Thus  the  total  pressure-head  in  feet  required  to  overcome 
inertia  and  the  hydraulic  resistances 


HYDRAULIC  ENGINE. 


and  is  represented  by  the  ordinate  between  the  parabola  ced 
and  the  line  ab  in  Fig.   199,  in  which  afgb  is  a  rectangle,  ab 

representing  the  stroke  2r, 

• 


the  pressure  due  to  inertia  at  the  end  of  the  stroke,  and 

oe  =  K— 


the  pressure  required  to  overcome  the  hydraulic  resistances  at 
the  centre  of  the  stroke. 


FIG.  199. 

The  ordinate  between  the  parabola  fmg  and  the  line  fg 
represents  the  back  pressure,  which  is  necessarily  proportional 

V'2 
to    the    square    of  the    piston-velocity,    i.e.,    to    -~^(r2  —  x*}. 

Hence  the  effective  pressure-head  on  the  piston,  transmitted 
to  the  crank-pin,  is  represented  by  the  ordinate  between  the 
curves  fmg  and  ced.  The  diagram  shows  that  the  pressure  at 
the  end  of  the  stroke  is  very  large  and  may  become  excessive. 
It  is  therefore  usual  to  introduce  relief-valves  or  air-vessels  ta 
prevent  violent  shocks.  In  certain  cases,  however,  as,  e.g., 
in  a  riveting-machine,  a  heavy  pressure  at  the  end  of  the 


LOSSES   OF  ENtKGY  4N  HYDRAULIC  ENGINES,  ETC.        35 r 

stroke,  just  where  it  is  most  needed  to  close  the  rivet,  is  of 
great  advantage,  and  therefore  the  inertia  effect  is  increased 
by  the  use  of  a  supply-pipe  of  small  diameter  and  an  accumu- 
lator with  a  small  water  section  (Fig.   197). 
By  equation  (i), 


This  speed  v  can  be  regulated  at  will  by  the  turning  of  a 
cock,  as  in  this  manner  the  hydraulic  resistances  may  be 
indefinitely  increased. 

Let  the  engine  be  working  steadily  under  a  pressure  P, 
and  let  VQ  be  the  speed  of  steady  motion.  Then 

and 

c        useful  resistance  overcome  by  the  piston 
(  -J-  friction  between  piston  and  accumulator-cylinder. 
If  P  is  diminished,  the  speed  V0  will  be  slightly  increased, 

but  in  no  case  can  it  exceed  \  /  --~~. 

V    wK 

5.  Losses  of  Energy. — The  losses  may  be  enumerated  as 
follows : 

(a)  The  Loss  Ll  due  to  Piston-friction. — It  may  be  assumed 
that  piston-friction  consumes  from  10  to  20  per  cent  of  the 
total  available  work. 

(V)  The  Loss  L2  due  to  Pipe-friction. — The  loss  of  head  in 
the  supply-pipe  of  diameter  d^ 

_  4//Q)2 
The  loss  of  head  in  the  discharge-pipe  of  diameter  dz 


35  2        LOSSES  OF  ENERGY  IN  HYDRAULIC  ENGINES,  ETC. 
Hence  the  total  loss  of  head  in  pipe-friction  is 


The  loss  in  the  relatively  short  working  cylinder  is  very 
small  and  may  be  disregarded. 

(c)  The  Loss  L3  due  to  Inertia.  —  The  work  expended  in 
moving  the  water  in  the  supply  -pipe 

_  wA     v*_ 

gn     7' 

and  in  moving  the  water  in  the  discharge-pipe 

wA      v* 
gn*  ,  2  ' 

The  total  work  thus  expended 


, 
'    nl2g 

and  it  may  be  assumed  that  nearly  the  whole  of  this  is  wasted. 
Hence  the  corresponding  loss  of  head  is 

_  wA_n     i'\f^_     w_[i_     /'W      ^ 

3  ~~  A2r\n    '    n'l2g  "  2r\n     '    n'l2g  ~    *2g' 

(d^    The  Loss  L4  due  to  Curves  and  Elbows.  —  The  losses 
due  to  curves  and  elbows  may  be  expressed  in  the  form 

Z4=/4J(Chap.  II,  Art.  14). 

(e]    The  Loss  LR  due  to  Sudden   Changes  of  Section.  —  The 
loss  of  head  in  the  passage  of  the  water  through  the  ports  may 

o 

be  expressed  in  the  form/'  —  . 

The  loss  occasioned  by  valves  may  also  be  expressed  by 

r^ 
f  27- 


HYDRAULIC  BRAKE.  353 

Thus  the  total  loss  is 

V1  2,2 


The  coefficient  y  may  be  given  any  desired  value  between 
o  and  oo  by  turning  a  valve,  so  that  any  excess  of  pressure 
may  be  destroyed  and  the  speed  regulated  at  will. 

(/")  The  Loss  L6  due  to  the  Velocity  with  which  the  Water 
leaves  the  Discharge-pipe. 


^  2g  *2g 

Hence 
the  effective  head  =  ^  -  (L^  +  L2  +  L3  +  L^+  L5  +  Z6), 

and  the  efficiency  ==  I  -  ^  +  L2  +  Z3  +  Z4  +  L5  +  Z6). 

/'o 

6.  Brakes.  —  Hydraulic  resistances  absorb  energy  which  is 
proportional  to  the  square  of  the  speed.  This  property  has 
been  taken  advantage  of  in  the  design  of  hydraulic  brakes  for 
arresting  the  motion  of  a  rapidly  moving  mass,  as  a  gun  or  a 
train,  of  weight  W.  In  Fig.  200  the  fluid  is  allowed  to  pass 

\  H 

a 


FIG.   200. 

from  one  side  of  the  piston  to  the  other  through  orifices  in  the 
piston. 

Let  m  be  the  ratio  of  the  area  of  the  piston  to  the  effective 
area  of  the  orifices. 

Let  v  be  the  velocity  of  the  piston  when  moving  under  a 
force  P. 

Let  A    be  the  sectional  area  of  the  cylinder. 


354  HYDRAULIC  BRAKE. 

Then 

the  work  done  per  second  =  Pv 

=  the  kinetic  energy  produced 

(m  —  I  )V 


and  therefore 

9 

P=  wA(m  —  O2  —  , 
'    2g 

which  is  the  force  required  to  overcome  the  hydraulic  resistance 
at  the  speed  v. 

Let  V  be  the  initial  value  of  7\  and  1\  the  maximum  value 
of  P.  Then 

1\  =  wA(m  -  i)2— 

>  2g 

Let  F  be  the  friction  of  the  slide.      Then 

V* 

P  +  F  =  wA(m  —  i)2—  +  F, 

and  P}  -f-  F  is  the  maximum  retarding  force.  It  would  cer- 
tainly be  an  advantage  if  tfhe  retarding  force  could  be  constant. 
In  order  that  this  might  be  the  case  (m  —  i)?>  must  be  con- 
stant, and  therefore  as  v  diminishes  m  should  increase  and 
consequently  the  orifice  area  diminish.  Various  devices  have 
been  adopted  to  produce  this  result. 

Assuming  the  retarding  force  to  be  constant,  let  x  be  the 
piston's  distance  from  the  end  of  the  stroke  when  its  velocity 
is  v.  Then 


and  therefore  T2  is  proportional  to  x. 
But  (in  —  i  )T;  is  constant. 
Therefore  (m  —  I  )  is  inversely  proportional  to    Vx. 


EXAMPLES.  355 


EXAMPLES. 

1.  A  4-ton  hydraulic  jack  with  a  2-in.  ram  and  a  i-in.  plunger  is  to 
lift  a  weight  of  i  ton,  and  is  worked  by  a  handle  with  a  leverage  of  12 
to  i.     If  the  efficiency  of  the   jack  is  80  per  cent,  what  force  must  be 
applied  to  the  handle  ?  Ans.  52^  Ibs. 

2.  The  ram  of  an  hydraulic  press  has  a  sectional  area-  50  times  as 
great  as  the  pump-plunger.     The  mechanical  advantage  of  the  lever  is 
10  to  i.     If  a  force  of  50  Ibs.  is  exerted  on  the  handle,  find  the  pressure 
on  the  ram.  Ans.  25,000  Ibs. 

3.  A  force  of  P  Ibs.  is  required  to  punch  a  hole  of  rtMns.  diameter. 
Firtd  the  diameter  of  the  ram,  the  available  fluid  pressure  being  p  Ibs 
per  square   inch.     If  this  pressure   is  developed  by  a  steam-intensifier 
with  a  steam-piston  area  n  times  that  of  the  intensifier's  ram,  find   the 

required  steam-pressure.  /~^LP      $ 

Ans.    \'  —  —  ;    £-. 
up      n 

4.  In  a  steel  hydraulic  press  the  fluid  pressure  is  6000  Ibs.  per  square 
inch,  and  the  maximum  allowable  stress  in  the  metal  is  18,000  Ibs.  per 
square  inch.     If  the  internal  diameter  of  the  press  is  12  ins.,  what  must 
the  thickness  of  the  metal  be  ?     If  the  thickness  of  the  metal  is  3  ins., 
what  must  the  internal  diameter  be?  Ans.  2.485  ins.;  14.485  ins. 

5.  A  straight-line  law  is  found  experimentally  to  connect  the  weight 
W  to  b'e  lifted  and  the  effort  E  on  the  handle.     Find  the  law  from  the 
following  data:  when   W  =  1605  Ibs.,  E  =  10  Ibs.,  and  when   W7  =  6805 
Ibs.,  E  =  $o  Ibs.     A  pressure-gauge  gives  the  fluid  pressure  as  1932  Ibs. 
per  square  inch,   when   W  =  7000  Ibs.  ;  find   the  frictional    loss  at  the 
leather,  and  if  there  is  the  same  percentage  of  loss  at  the   two  leathers 
find  the  law  connecting  E  and  the  force  P  on  the  plunger.     The  experi- 
ments were  made  on  a  jack  with  a  2^-in.  ram,  a  f-in.  plunger,  and  a 
lever  with  a  velocity  ratio  of  30.      (Perry.) 

Ans.   W  =  305  +  130  E;  9.1  per  cent;  P  =  41  +17.5  E. 
(Perry's  "  Applied  Mechanics.") 

6.  An  accumulator-ram  is  8.8  ins.  in  diameter  and  has  a  stroke   of 
21  ft.       Find  the  store  of  energy  in  foot-pounds  when  the  ram  is  at  the 
top  of  its  stroke  and  is  loaded  till  the  pressure  is  750  Ibs.  per  square 
inch.  Ans.  958,320  ft. -Ibs. 

7.  In  a  differential  accumulator  the  diameters  of  the  spindle  are  7  ins. 
and  5  ins  ;  the  stroke  is  10  ft.     Find  the  store  of  energy  when  full  and 
loaded  to  2000  Ibs.  per  square  inch.  Ans.  377,000  ft. -Ibs. 

8.  The  pressure  on  a  5-in.  ram  is  to  be  1000  Ibs.  per  square  inch,  and 


356  EXAMPLES. 

the  supply  comes  from  a  tank  100  ft.  high.    Find  the  necessary  diameter 
of  the  piston  in  the  intensifier.  Ans.  24  ins. 

9.  In  a  differential  press  the  diameters  of  the  upper  and  lower  portions 
of  the  ram  are  6  ins.  and  8  ins.  respectively.     The  pressure  is  1000  Ibs. 
per  square  inch,  and  the  stroke  is  10  ft.     Find  the  load  on  the  accumu- 
lator, the  maximum  store  of  energy,  and  the  store  of  water. 

Ans.  22,000  Ibs. ;  220,000  ft. -Ibs. ;  i\\  cu.  ft. 

10.  What  load  must  be  applied  to  a  differential  accumulator  to  give 
a  pressure  of  1600  Ibs.  per  square  inch  ?   The  upper  and  lower  diame- 
ters of  the  ram  are  3  and  3!  ins.  respectively,  and  the  friction  of  the  cup- 
leathers  may  be  taken  as  5  per  cent  of  the  gross  load. 

Ans.  6062  Ibs.  ;  6700  Ibs. 

11.  Find  the  weight   which    will  give  an  average  fluid  pressure  of 
750  Ibs.  per  square  inch  in    an   accumulator  with   a  14-in.  ram  and  a 
stroke  of  16  ft.     How  much  energy  can  be  stored  up  ?    Find  the  friction 
at  each  cup-leather,  assuming  that  between  slow  rising  and  falling  the 
pressure  fluctuates  between  780  and  738  Ibs.  per  square  inch.     If  tfce 
pressure  is  750  Ibs.  per  square  inch  at  mid-lift,  find  the  actual  fluctua- 
tion. Ans.  1 15,500  Ibs. ;  1,848,000  ft.-lbs. ;  3234  Ibs.  ;  3769  Ibs. 

12.  An  accumulator,  loaded  to  a  pressure  of  750  Ibs.  per  square  inch, 
has  a  ram  of  21  ins.  diameter,  with  a  stroke  of  24  ft.     How  much  H.P. 
can  be  obtained  for  a  period  of  50  seconds  ?  Ans.  226.8. 

13.  An  accumulator  under  a  load  of  200,000  Ibs.  is  to  transmit  100 
H.P.  through  a4-in.  pipe  i  mile  long  with  a  loss  of  10  per  cent.    What 
should  be  the  diameter  of  the  ram,  the  coefficient  of  pipe  friction  being 
.006  ?  Ans.  17.33  '11S- 

14.  A  steam-accumulator  has  to  develop  a  total  force  of  66,000  Ibs. 
upon  the  ram  of  a  punch.     The  piston  area  is  15  times  that  of  the  hy- 
draulic-cylinder, which  has  a  diameter  of  10  inches.    Find  the  intensities 
of  the  steam  and  the  water-pressure,.  Ans.  56  Ibs.  ;  840  Ibs. 

15.  The  piston  and    ram   areas    of  a  steam-accumulator  are  in  the 
ratio  of  10  to  i.    'Find  their  diameters  so  that  a  steam-pressure  of  100 
Ibs.  per  sq.  in.  may  develop  a  total  load  on  the  ram  of  38,500  Ibs. 

Ans.  22.136  ins. ;  7  ins. 

16.  A  Brotherhood  engine  with  a  4-in.  cylinder  and  a  3-in.  stroke 
makes  50  revols.  per  minute.     The  average  motive  pressure  is  700  Ibs. 
per  sq.  in.,  and  the  average  back  pressure,  due  to  frictional  resistances, 
etc.,  is  210  Ibs.  per  sq.  inch.     Find  the  H.P.  developed,  and  also  deter- 
mine the  diameter  of  the  cylinder  if  only  one  half  o>i  this  power  is  to  be 
developed.  Ans.  7  ;  2.83  ins. 

17.  A  crane   with   an  hydraulic    efficiency  of   .9  and    a   mechanical 
efficiency  of  .45  is  worked  by  water  at  a  pressure   of  750  Ibs.  per  sq. 
inch.     The  piston  has  an  effective  area  of  96  sq.  ins.  on  one  side,  48 
sq.  ins.  on  the  other,  and  pushes  a  three-sheave  pulley-block.     Find  the 
maximum  weight  which  can  be  lifted  and  the  work  done  per  gallon  of 


EXAMPLES.  357 

water,  first  when  the  water  presses  on  one  side  only,  and  second  when  it 
presses  on  both  sides.  Also  find  the  work  done  per  gallon  of  water 
when  the  full  loads  in  the  two  kinds  of  working  are  being  lifted. 

Ans.  4860  Ibs.  ;  6998.4  ft.-lbs.;  2430  Ibs.;  3499.2  ft.-lbs.  ;  6998.4 
ft.-lbs. 

18.  An  hydraulic  crane  with  a  velocity  ratio  of  9  and  a  mechanical 
efficiency  of  .75  has  to  lift  a  weight  of  10,000  Ibs.     It  is  worked  by  water 
at  a  pressure  of  750  Ibs.  per  sq.  in.,  and  the  frictional  loss  of  pressure  is 
91  Ibs.  per  sq.  inch.     Find  the  diameter  of  the  ram.          Ans.  15.2  ins. 

19.  The  two  wire  ropes  from  the  cage  of  a  ram-lift  pass  vertically 
over  a  pulley  to  a  counterweight,  and  the  ram  rises  from  100  ft.  below  to 
20  ft.  above  the  level  of  the  supply-pipe.     Water-pressures  of  500  ibs. 
and  loo  Ibs.  per  sq.  in.  act  upon  a  3|-m.  and  a  7-in.  ram,  respectively. 
Find  the  weight  of  the  ropes  per  lineal  foot  and  the  lifting  force  at  the 
top  and  bottom  of  the  stroke. 

Ans.  4.2  Ibs.,  16.7  Ibs.  ;  5230  Ibs.,  4729  Ibs.  ;  55.21  Ibs.,  3516  Ibs. 

20.  Find  the  pressure  due  to  inertia  at  the  end  of  the  out-stroke  of  a 
rotary  motor  with  a  4-m.  piston  and  a  7-in.  stroke,  driven  by  water  in  a 
4-111.  supply-pipe  250  tt.  long.     The  motor  makes  125  revols.  per  minute, 
and  the  length  of  the  connecting-rod  is  15  inches. 

Ans.  20.7  Ibs.;  12.9  Ibs. 

21.  A  direct-acting  lift  has  a  ram  9  inches  diameter,  and  works  under 
a  constant  head  of  73  feet,  of  which  13  per  cent  is  required  by  ram  fric- 
tion and  friction  of  mechanism.     The  supply-pipe  is  100  feet  long  and  4 
inches  diameter.     Find  the  speed  of  steady  motion  when  raising  a  load 
of  1350  Ibs.,  and  also  the  load   it  would  raise  at  double  that  speed. 
(f  =  .00672.) 

If  a  vaive  in  the  supply-pipe  is  partially  closed  so  as  to   increase  the 
coefficient  of  resistance  by  5$,  what  would  the  speed  be? 

Ans.  Speed  =  2  ft.  per  second  ;  load  =  150  Ibs. 

22.  Eight  cwt.  of  ore  is  to  be  raised  from  a  mine  at  the  rate  of  900 
feet  per  minute  by  a  water-pressure  engine,  which  has  four  single-acting 
cylinders,  6  inches  diameter,    18  inches  stroke,  making  60  revolutions 
per  minute.    Find  the  diameter  of  a  supply-pipe  230  feet  long  for  a  head 
of  230  feet,  disregarding  resistances  and  taking/  =  .006. 

Ans.  Diameter  =  4  inches. 

23.  If  A.  be  the  length  equivalent  to  the  inertia  of  a  water-pressure 
engine,  F  the  coefficient  of  hydraulic  resistance,  both  reduced  to  the 
ram,  z/0  the  speed  of  steady  motion,  find  the  velocity  of  ram  after  moving 
from  rest  through  a  space  x  against  a  constant  useful  resistance.     Also 
find  the  time  occupied. 


24.  An  hydraulic  motor  is  driven  from  an  accumulator,  the  pressure 


35 8  EXAMPLES. 

in  which  is  750  Ibs.  per  square  inch,  by  means  of  a  supply-pipe  900  feet 
long,  4  inches  diameter;  what  would  be  the  maximum  power  theoreti- 
cally attainable,  and  what  would  be  the  velocity  in  the  pipe  correspond- 
ing to  that  power  ?  Find  approximately  the  efficiency  of  transmission 
at  half  power,/  =  .007. 

Ans.  H.P.  =  250;  v  =  22  ft.;  efficiency  =  .66  nearly. 

25.  A  gun  recoils  with  a  maximum  velocity  of  10  feet  per  second. 
The  area  of  the  orifices  in  the  compressor,  after  allowing  for  contraction, 
may  be  taken  as  one  twentieth  the  area  of  the  piston.  Find  the  initial 
pressure  in  the  compressor  in  feet  of  liquid. 

Assuming  the  weight  of  the  gun  to  be  12  tons,  friction  of  slide  3 
tons,  diameter  of  compressor  6  inches,  fluid  in  compressor  water,  find 
the  recoil. 

Find  the  mean  resistance  to  recoil.  Compare  the  maximum  and 
mean  resistances,  each  exclusive  of  friction  of  slide. 

Ans.  621 ;  4  ft.  2$  in. ;  total  mean  resistance  =  4.4  tons;  ratio  =  2.5. 


CHAPTER    V. 


IMPACT,  REACTION,  IMPACT   AND   TANGENTIAL 
TURBINES. 

NOTE. — The  following  symbols  are  used: 

vl  =  the  velocity  of  the  jet  before  impact ; 

i>z  —     ' '        "          i « .  i «    « i   after  leaving  the  vane ; 
u  =     "        "          "    "   vane; 

F=     "        "          "     "   water  relatively  to  the  vane; 

A  ==  sectional  area  of  the  impinging  jet ; 

m  =  mass  of  the  water  reaching  the  vane  per  second. 

i.  Impact  of  a  Jet  upon  a  Flat  Vane  Oblique  to  the 
Direction  of  the  Jet. — Let  6  be  the  angle  between  the  normal 
to  the  vane  and  the  direction  of  the  impinging  jet,  0  the  angle 
between  the  normal  to 
the  vane  and  the  direc- 
tion of  the  vane's  mo- 
tion, and  a  the  angle 
between  the  vane  and 
the  vertical. 

The  jet,  moving 
with  its  stream-lines 
parallel,  swells  out 
near  the  vane,  over 
which  it  spreads  and 
with  which  it  travels 
along  in  the  direction 
of  the  vane's  motion,  and  finally  again  flows  along  with  its 
stream-lines  sensibly  parallel  to  the  vane. 

359 


360  IMPACT  ON  FLAT   YANES. 

The  problem  is  still  further  complicated  by  the  production 
of  eddies  and  vortices  for  which  allowance  can  only  be  made 
in  a  purely  empirical  manner. 

Let  N  be  the  normal  pressure  on  the  vane  due  to  the 
impact. 

Let  N'  be  the  total  normal  pressure  on  the  vane. 

Let  J'Fbe  the  weight  of  water  on  the^vane. 

Then 

N  —  N'  —  -  W  sin   a  =  change  of  momentum  in  direction  of 

the  normal 

=  mi\  cos  0  —  mu  cos  0, 
or 

N  =  m(vl  cos  6  —  u  cos  0),       ...      (i) 

(N.B.  The  sign  in  front  of  u  cos  0  will  be  plus  if  the  jet 
and  vane  move  in  opposite  directions.) 

The  term  W  sin  a  maybe  designated  the  static  pressure, 
and  the  term  m(vl  cos  6  —  u  cos  0)  the  dynamic  pressure, 
which  causes  the  deviation  of  the  stream-lines. 

NOTE.  —  The  pressure  when  a  jet_/?r^/  strikes  the  plane  is 
greater  than  when  the  flow  has  become  steady,  or  a  permanent 
regime  is  established. 

This  is  made  evident  by  the  following  consideration: 

At  any  moment  let  MN,  PQ,  *RS  be  the  bounding  planes 
across  which  the  water  is  flowing  with  its  stream-lines  sensibly 
parallel. 

In  a  unit  of  time  let  the  bounding  planes  of  the  mass  be 
M'N',  P'Q',  R'S'. 

Then,  initially,  the  reaction  of  the  plane  must  destroy  the 
motion  of  the  mass  of  the  fluid  bounded  by  M'N'  ,  P  'Q  ',  and 
R'S'. 

Take  OC  to  represent  i\  in  direction  and  magnitude. 


In  one  second  the  vane  AB  moves  parallel  to  itself  into  the 
position  A'B'.      Let  A'B'  intersect  OC  in  D. 


IMPACT  ON  FLAT   k,1NES.  301 

Then 


m  =  -A  .  DC  =  -A(v,  -  OD) 
g  g  ' 


W      I  COS   0\ 

=  -A(v.  —  u—    ~)  .......     (2) 

\  l         cos  Qi 


. 
g     \  l         cos  Q 

Thus  equation  (i)  becomes 

w    A 

N  =  -       —  z(v,  cos  6  -  u  cos  0)2.       ...     (3) 
g  cos  6^ 

Let  P  be  the  pressure  in  the  direction  of  the  vane's  motion. 
Then 

w     cos  0 
/>  =  A^  cos  0  =  -A  ^-£  (v,  cos  0  -  u  cos  0)2,    .      (4) 


and  the  useful  work  done  on  the  vane  per  second 

—A 
g     cos 


=  Pu  =  —A  ---  ,-,#(7',  cos  0  —  u  cos  0)2.  (5) 

cos  0 


W      V  3 

The  total  available  work  =  -A—  .....  (6) 

g      2 

W       COS  0 

—^4  -       ^,2/(>i    COS    G  —  U  COS  0)2 

•LJ  4.u         ^r    •  J?"        cos  ^ 

Hence  the  efficiency  •=.  e  __ 


COS  0    W 
=    2C^ST^  3^1  C°S 

This  is  a  maximum  when 

vl  cos  ^  =  3//  cos  0,      .....      (8) 
and  therefore 

o 

the  maximum  efficiency  =  —  cos2  6.       .      .      (9) 

If  the  vane  is  of  small  sectional  area,  a  portion  of  the  water 
will  escape  over  the  boundary  and  the  pressure  must  necessarily 
be  less  than  that  given  by  equation  (3). 


362  IMPACT  ON  FLAT   YANES. 

Series  of  Vanes. — Instead  of  one  vane  moving  before  the 
jet,  let  a  series  of  vanes  be  introduced  at  short  intervals  at  the 
same  point  in  the  path  of  the  jet. 

The  quantity  of  water  now  reaching  the  vane  per  second 
is  evidently 

w 
m=-Avl9 (10) 

o 

and,  by  equation  (i),  the  normal  pressure 

w 
N  •=.  ~Avl(vl  cos  V  —  u  cos  0).   .      .     .      (11) 

o 

Also,  the  pressure  in  the  direction  of  the  motion  of  the  vane 

=  P  —  A^cos  0  = — Avl(vl  cos  6  —  u  cos  0)  cos  0.        (12) 
<s 

The  useful  work  done  per  second 


IV 

=  Pu  =  —  Avlu(vl  cos  6  —  u  cos  0)  cos  0,      .     (13) 

o 


and  the  efficiency 


—  Avlu(vl  cos  6  —  u  cos  0)  cos  0 
«,     £ 


1  cos  6  —  u  cos  0)  cos  0 

~~ 


This  is  a  maximum  when  z/j  cos  6  =  2u  cos  0,   .     .     (15) 
and  therefore 

cos*  8 
the  maximum  efficiency  —  —  -  —  .   .      .      .     (16) 


EXAMPLES.  363 

Ex.    i.     Let  a  single  vane   be  at  right  angles  to,  and  move  in  the 
line  of,  the  jet's  motion,  Fig.  202. 
Then  6  =  o  =  <£.     Hence 

"W 

the  pressure  —P  --  N  =  -  A(v*  —  «)3;       ....     (17) 

the  useful  work    =  Pu  =  —Au(v\  —  uY;     .     .     .     .     (18) 
g 

K 

the  efficiency  =  2—(vi  —  u)^;       ....     (IQ> 

FIG.  202.  l 

the  maximum  efficiency^  — (20) 

Again,  if  u  =  o,  i.e.,  if  the  vane  be  fixed,  and  if  H  be  the  head  corre- 
sponding to  the  velocity  7/1,  then,  by  equation  (17), 

P  —  —  Av?  —  2wAH 

g 
=  twice  the    weight  of  a   column    of   water 

of  height  //and  sectional  area  A. 

Ex.  2.  Let  each  of  a  series  of  vanes  be  at  right  angles  to,  and  move 
in  the  line  of,  the  jet's  motion  at  the  instant  of  impact. 
Then  6  =  o  =  0.     Hence 

the  pressure  —  N  =  P  =  —Av*(vi  —  u);       .     .     .     .     .     (21) 
the  useful  work  =  Pu      =  —  Av\u(y\  —  u)\ (22) 

& 


,  a-     •  2u(Vi—u] 

the  efficiency  =  —  ; ; 


(23) 


the  maximum  efficiency  =  — (24) 

Ex.3.  A  stream  of  .125  sq.  ft.  sectional  area  delivers  locu.  ft.  of  water 
per  second  and  impinges  normally  against  a  flat  vane.  It  is  required  to 
find  (a)  the  pressure  on  the  vane  if  fixed ;  (ff)  the  pressure  and  the  useful 
effect  if  the  vane  moves  in  the  direction  of  the  jet's  motion  with  a 
velocity  of  40  ft.  per  second  ;  (<r)  the  pressure  and  useful  effect  when  the 
single  vane  in  (b)  is  replaced  by  a  series  of  vanes  which  follow  each 
other  at  intervals  of  a  second. 

The  velocity  of  the  jet  before  impact  =  -  -  =  80  ft.  per.  sec. 

(a)  The  pressure  on  vane  =  momentum  of  jet  =  — '-?  x  lox  80=1562^  Ibs. 
(£)  The  quantity  of  water  reaching  the  vane  per  sec. 
=  -77(80  —  40)  =  5  cu.  ft. 


364  IMPACT  ON  SURFACES   OF  RESOLUTION. 

The  pressure  on  the  vane  =  momentum  of  jet 

=      *5(8o  -40)  =  390*  Ibs. 


The  useful  effect  =  390!  x  40  =  15,625  ft.-lbs. 

The  total  available  work  =  —  -10  .  —  =  62,500  ft.-lbs. 

Therefore  the  efficiency  =  -4  —  3  =  --. 
62500       4 

\c)  The  quantity  of  water  now  reaching  the  vane  per  second 

=  -  x  80  —  10  cu.  ft. 

o 

The  pressure  on  the  vane  =  momentum  of  jet 

=  _?iIO(8o  —  40)  =  78  ii  Ibs. 
The  useful  effect  =  781^  x  40  =  31,250  ft.-lbs. 
The  efficiency  =  ,  --  =  —  . 

y          62500         2 

Ex.  4.  The  jet  in  the  preceding  example  impinges  upon  a  vane  with 
its  normal  inclined  at  60°  to  the  jet's  direction,  and  is  driven  with  a 
velocity  of  20  ft.  per  second  in  a  direction  making  an  angle  of  30°  with 
the  vane's  normal.  Find  (a)  the  pressure  on  the  vane  ;  (b}  the  useful 
effect. 

(a)  The  quantity  of  water  reaching  the  jet  per  second 

l\  =  |(4  -  i/3)  =  5-67  cu.  ft. 
The  relative  velocity  in  the  direction  of  the  normal 

=  80  cos  60°  —  20  cos  30°  =  10(4  —  V  3)  =  22.68  ft.  per  sec. 

The  normal  pressure  upon  the  vane  =  momentum    in   direction   of 

normal 

=  5.67  x  22.68  =  128.6  Ibs. 

The  pressure  in  direction  of  vane's  motion  =  128.6  cos  30° 

=  111.35  Ibs. 

(b)  The  useful  effect  =  111.35  x  2O  =  2227  ft.-lbs. 

•9  2^7 

The  efficiency  =  ^  —  ^-  =  .0356. 
3       625000 

5.  Jet  of  Water  Impinging  upon  a  Surface  of  Revolution 
Moving  in  the  Direction  of  its  Axis  and  also  in  the  Line  of 
the  Jet's  Motion.  —  The  relative  velocity  of  the  jet  is  i\  —  u 

if  the  jet  and   surface   move  in  the  same  direction,  Figs.  203 
and  204,  and  z^  -j-  u  if  they  move  in  opposite  directions,  Figs. 


i  /  cos 

—  _(  80  —  20  - 
8V  cos  60°  i 


IMPACT  ON  SURFACES  OF  RESOLUTION. 


365 


205  and  506.      This  relative  velocity,  if  friction  is  disregarded, 
'.remains  unchanged  in  magnitude  as  the  water  flows  over  the 
surface,  but  the  stream-line  direction  is  deviated  through  an 
angle  /?. 


FIG.  205.  FIG.  206. 

Let  the  water  leave  the  surface  at  D,  and  in  the  direction 
of  the  tangent  at  D  take  DE  =  v^  —  #,  Figs.  203  and  204, 
and  DE  =  v^  -\-  u,  Figs.  205  and  206.  Draw  DF  parallel 
to  the  axis  and  take  DF  =  u. 

Complete  the  parallelogram  DEGF. 

Then  DG,  the  diagonal,  must  represent,  in  direction  and 
magnitude,  the  absolute  velocity,  v2 ,  with  which  the  water 
leaves  the  surface. 

Hence,  from  Figs.  203  and  204, 

V*  =   U*  +  (?Y-  U?  -  2U(V1  -   U)  COS   (1 80°  -  /?),     .        (I) 

and  the  work  done  by  the  water  on  the  surface 
v*  -  v* 


(2) 


=  mu(i\  —  u)(i  —  cos  /?) 
wA  6 


366  IMPACT  ON  SURFACES   OF  RESOLUTION. 

From  Figs.  205  and  206, 

v£  =  u*  -\-  (^  -|-  u)2  —  2«(z/j  +  «)  cos  fi, 
and  the  work  done  by  the  surface  on  the  water 


+  «)(l  —  cos  /?) 
u(vt  +  u)2  sin2     .....     (3) 


Let  /*  be  the  pressure  on  the  surface  in  the   direction  of  its 
motion.      Then 

Pu  =  work  done  =  2  -  u(vl  =£  u)2  sin2  —  , 

and  therefore 

P=  2—  (vqpu)2  sin2-  ........     (4) 

6 

The  efficiency  for  the  case  of  Figs.  203  and  204 


2  , 

2  —  u(v,  —  uY  sin"2—  A.U(V,  —  uY  sin4— 

^                                    2  ^    l            ^             2 

~  ~~ 


g    2 

16         /? 
which  is  a  maximum  and  =  —  sin2  —  when  vl  =  $u. 

Series  of  Surfaces.  —  If  a  number  of  surfaces  are  successively 
introduced  at  short  intervals  at  the  same  point  in  the  path  of 
the  jet,  the  quantity  of  water  reaching  each  surface  per  second 
becomes 

wA 
m  =  ----  v,. 


IMPACT  ON  SURFACES   OF  RESOLUTION.  367 

In  this  case 

wA  /5 

the  work  done  =  2 — v^Vj  qp  u)  sin2- ,   .     .     (6) 

wA  ft 

and  the  pressure  =  2 —  v^Vj  qp  u)  sin2  - .      .     .     (7) 

Also,  the  efficiency,  when  the  water  drives  the  surface, 
w  A       /  .   2ft 

*  fWl         ")  Sm   2 


wA  v 


/? 

—  «    sin2  - 


which  is  a  maximum  and  ==  sin2  —  when  v^  =  2u. 

With    a    convex     surface     /?  <  90°,    and     the     coefficient 

ft 

2  sin2  — ,  or  I  —  cos      ,  is  less  than  unity. 

With     a    concave     surface    f3  >  90°,    and    the    coefficient 

/? 

2  sin2  -,  or   I  —  cos  /3,  is  greater  than  unity. 

If  the  surface  be  of  the  cup  type  and  hemispherical,  the 

1 80° 
maximum    efficiency  =  sin2  =  I,   since  j3  =  180°.      The 

water  should  therefore  leave  the  surface  without  velocity,  and, 
substituting  ^  =  2u  and  ft  =  180°  in  equation  (i), 

v£  =  M2  _[_  (2u  —  lif  —  2U(2U  —  11}  =  O. 

Ex.  A  jet  of  water  of  .125  sq.  ft.  sectional  area  delivers  12  cu.  ft.  of 
water  and  impinges  axially  upon  a  120°  cone.  Find  (a)  the  pressure  on 
the  cone  when  fixed,  and  (b)  the  pressure  on  the  cone  and  the  useful 


368  IMPACT  ON  BORDERED    YANE. 

effect  when  the  cone  is  driven  in  the  direction  of  its  axis  with  a  velocity 
of  32  ft.  per  second. 

The  velocity  of  the  jet  before  impact  =  -^  =  96  ft.  per  sec. 

62i  ,  60° 

(a)  Pressure  on  convex  surface  =      2  — ^  .  12.96  sma  —  =  1125  IDS. 

Pressure  on  concave  surface  =  2—12.96  sin2  —  -  =  3375  Ibs. 

(b)  When  the  water  impinges  on  the  convex  surface 

the  work  done  =  2 — --^32(96  —  32)"  sin2  —  =  16,000  ft.-lbs., 

16000 
the  pressure      =    =  500  Ibs. 

*) 

When  the  water  impinges  on  the  concave  surface 

the  work  done  =  2—  -£-32(96  —  32)2  sin2  ^-  =  48,000  ft.-lbs., 

32    5  2 

A8000 

the  pressure       = =  1500  Ibs. 

6.  Impact  of  a  Jet  of  Water  upon  a  Vane  with  Borders. 

— Let  the  vane  in  Art.  I  be  provided  with  borders,  Figs.  207 
and  208,  so  as  to  produce  a  further  deviation  of  the  stream- 
lines, and  let  the  water  finally  flow  off  with  a  velocity  v2  in  a 
direction  making  an  angle  0'  with  the  normal  to  the  vane. 


FIG.  207.  FIG.  208. 

Then 
the  normal  pressure  =  N 

=  mvl  cos  0  =F  mv^  cos  0'  ^p  mu  cos  0 
—  m(vt  cos  0  T  ?'2  cos  6'  ^f  u  cos  0), 

the  sign  of  the  second  term  being  plus  or  minus  according  to 
the  direction  in  which  the  stream-lines   are  finally  deviated. 


IMPACT  APPARATUS.  369 

The  effect  of  the  borders  is  therefore  to  increase  or  diminish 
tfre  normal  pressure,  and  hence  also  the  useful  work  and  the 
efficiency. 

SPECIAL  CASE.  —  Let  the  vane  be  at  rest,  i.e.,  let  u  =  o, 
and  let  the  final  and  initial  directions  of  the  jet  be  parallel. 

Also,  let  vl  =  vr     Then 

N  =  m(vl  cos  6  +  vl  cos  6) 

w 

=  2— AV?  cos  e 
g 

cos  6. 

Hence,  if  6  =  0,  the  normal  pressure  N  —  ^.wAH  —  four 
times  the  weight  of  a  column  of  water  of  height  H  and  sec- 
tional area  A. 

7.  Impact  Apparatus  in  Hydraulic  Laboratory,  McGill 
University. — This  apparatus  was  constructed  for  the  purpose 
of  determining  the  force  with  which  jets  from  orifices,  nozzles, 
etc.,  impinge  upon  vanes  of  different  forms  and  sizes. 

A  massive  cast-iron  bracket,  Fig.  209,  has  one  end  securely 
bolted  to  the  front  of  the  tank,  and  the  other  supported  by  a 
vertical  tie-rod  from  one  of  the  oak  beams  in  the  ceiling.  The 
upper  surface  is  provided  with  accurately  planed  slides,  which 
are  set  level  about  5  ft.  above  the  orifice  axis.  If,  from  any 
cause,  the  end  of  the  bracket  farthest  from  the  tank  is  found  to 
be  too  high  or  too  low,  the  error  can  be  corrected  by  loosen- 
ing or  tightening  the  nut  on  the  tie-rod. 

The  balance  proper  is  carried  by  a  sliding  frame  which  can 
be  moved  horizontally  into  any  position  along  the  bracket  by 
means  of  a  rack  and  pinion  actuated  by  a  sprocket-wheel  with 
chain.  At  one  end  the  frame  has  two  equal  arms  with  a 
common  horizontal  axis  parallel  to  the  bracket,  and  each  arm 
has  a  stop  on  its  lower  surface  which  serves  to  limit  the  oscil- 
lation of  the  balance. 


37° 


IMPACT  APPARATUS. 


The  balance,  in  its  mean  position,  consists  of  a  main  trunk 
with  horizontal  axis  rigidly  connected  with  a  vertical  slotted 
arm  and  with  two  equal  horizontal  arms  at  one  end.  The 
common  axis  of  the  latter  is  horizontal  and  perpendicular  to 
the  axis  of  the  main  trunk.  The  hardened-steel  knife-edges 
of  the  balance  are  4  ft.  centre  to  centre  and  rest  in  hardened- 
steel  vees  inserted  in  the  ends  of  the  sliding  frame  on  each 
side  of  the  bracket.  The  bottom  of  each  vee  is  in  the  same 


FIG.  209. 

horizontal  line  (called  the  axis  of  the  vees)  at  right  angles  to 
the  bracket. 

A  bar  with  the  upper  portion  graduated  in  inches  and 
tenths  has  a  slot  in  the  lower  portion,  which  is  bent  into  a 
circular  segment  of  9^  ins.  radius.  The  bar  slides  along  the 
slot  in  the  vertical  arm  of  the  balance.  A  radial  block,  with 
the  holder  into  which  the  several  vanes  are  screwed,  moves 
along  the  slot  in  the  circular  segment,  and  may  be  clamped  in 
any  required  position,  the  angular  deviations  from  the  vertical 


COEFFICIENT  OF  IMPACT.  37 1 

being  shown  by  graduations  on  the  segment.  The  centre  of 
this  segment  in  every  case  coincides  with  the  central  point  of 
impact  on  a  vane,  is  in  the  vertical  axis  of  the  balance-arm, 
and  is  also  vertically  below  the  axis  of  the  vees.  Thus  the 
jet  can  always  be  made  to  strike  the  vane  both  centrally  and 
normally. 

The  scale-pan  hangs  from  a  knife-edge  at  one  end  of  the 
horizontal  arms  of  the  balance,  while  to  the  other  end  is 
attached  a  fine  pointer,  which  indicates  the  angular  movement 
of  the  balance  on  a  graduated  arc  fixed  to  the  sliding  frame. 
The  balance  is  in  its  mid-position  when  the  pointer  is  opposite 
the  zero  mark. 

When  a  vane  has  been  secured  in  any  given  position,  the 
preliminary  adjustment  of  the  balance  is  effected  by  moving  a 
heavy  cast-iron  disc  along  a  horizontal  screw  fixed  into  the 
main  trunk.  The  sensitiveness  of  the  balance  is  also  increased 
or  diminished  by  raising  or  lowering  heavy  weights  on  two 
vertical  screws  in  the  top  of  the  trunk. 

Assume  that  the  adjustments  have  all  been  made  and  that 
the    jet,    Fig.    210,    now    impinges 
normally  upon  a  vane. 

Let  Wbe  the  weight  required  in 
the  scale-pan  to  bring  the  balance 
back  into  its  mid-position. 

Let  Fa  be  the  actual  force  of  im- 
pact determined  by  the  balance. 

Let  Ft  be  the  theoretical  force  of 
impact  deduced  by  the  ordinary 
formulae. 

Fa 

Then  the  ratio    =£  =  c£  may  be  called  the    coefficient  of 
r  t 

impact. 

Let  y  be  the  vertical  distance  of  the  central  point  of  impact 
below  the  horizontal  axis  of  the  orifice,  which  is  36  ins.  below 


w 


372  COEFFICIENT  OF  IMPACT. 

the  axis  of  the  vees.  The  distance  between  this  axis  and  the 
point  of  suspension  of  the  scale-pan  is  24  ins. 

Let  v  be  the  velocity  with  which  the  water  issues  from  the 
orifice. 

Let  v'  be  the  velocity  of  the  jet  at  the  point  of  impact. 

Then 


Q  being  the  delivery  per   second  and  ft  the  angle   through 
which  the  water  is  turned  on  the  vane. 

If  the  axis  of  the  jet  at  the  point  of  impact  makes  an  angle 
6  with  the  horizontal,  then 

v'  cos  6  =  v  =  cv  V2gh. 

Therefore 

tv  /3 

Ft  cos  0  =  2~~Qv  sin2—. 

Again,  taking  moments  about  D, 

Fa  cos  #(36+7)  =   W.  24. 
Hence 

F 


ft 
6W 


ft' 
wc<fcz;2Ah(36  +  y)  sin2- 


A  being  the  sectional  area  of  the  orifice. 

A  large  number  of  experiments  have  been  made  for  the 
purpose  of  determining  the  value  of  c{  and  are  described  in  the 
Trans,  of  the  Royal  Soc.  of  Can.,  Vol.  II,  1896,  and  of  the 
Can.  Soc.  of  Civil  Engineers,  Vol.  XII.  No  definite  law  of 


REACTION.  373 

variation  has  yet  been  found,  but  the  following  general  results 
have  been  obtained: 

The  actual  force  of  impact  is  always  much  less  than  that 
indicated  by  theory.  Even  under  the  most  favorable  condi- 
tions, with  a  very  large  coefficient  of  velocity,  the  theoretical 
force  of  impact  was  found  to  exceed  the  actual  by  3  or  4  per 
cent. 

The  coefficient  of  impact,  cf ,  increases  with  the  velocity  of 
the  jet. 

The  coefficient  rapidly  diminishes  with  the  angle  through 
which  the  stream  is  deflected.  It  is  also  of  interest  to  note 
that,  with  small  angles  of  deflection,  ct  was  greatest  with  a 
concave  parabolic  vane,  less  with  an  elliptic,  and  least  with  a 
circular,  but  that  this  order  was  reversed  when  the  deflections 
were  larger. 

8.  Reaction — Jet  Propeller. — The  term  reaction  is  em- 
ployed to  denote  the  pressure  upon  a  surface  due  to  the  direc- 
tion and  velocity  with  which  the  water  leaves  the  surface. 
Water,  for  example,  issues  under  the 
head  h  and  with  the  velocity  vl  (at  con- 
tracted section)  from  an  orifice  of  sectional 
area  A  in  the  vertical  side  of  a  vessel,  ~^j  f  ^ 
Fig.  211. 

Let  R  be  the  reaction  on  the  opposite 

vertical  side  of  the  vessel,  and  let  Q  be  the  quantity  of  water 
which  flows  through  the  orifice  per  second.      Then 

R  —  horizontal  change  of  momentum 

wQ          w 


.     .      (i) 

&  <D 

disregarding  the  contraction  and  putting  cv  =  I. 

Thus  the  reaction  is  double  the  corresponding  pressure 
when  the  orifice  is  closed  (Ex.  i,  p.  363). 

Again,  let  the  vessel  be  propelled  in  the  opposite  direction 
with  a  velocity  u  relatively  to  the  earth. 


374  REACTION. 

Then  v^  —  u.  is  the  velocity  of  the  jet  at  the  contracted 
section  relatively  to  the  earth  and 

R  =  horizontal  change  of  momentum 

w 

=  -fi("i-«) •  •  (2) 

The  useful  work  done  by  the  jet 

w. 

The  energy  carried  away  by  the  issuing  water 

w(v,  -  uY 


Hence 


w                           w    (v  — 
the  total  energy  =  ~Qu(i\  —  u)  -\ Q>~ 

O  O  •" 


W       V 

-'--,    ......     (5) 


and 


w 

-Qu(y^  -  u] 
the  efficiency  2u 


IV      V*   —   UZ          ^  +  U 
g  2~" 

Thus  the  more  nearly  v^  is  equal  to  uy  and  therefore  the 
larger  the  area  A  of  the  orifice,  the  greater  is  the  efficiency. 

If  the  vessel  is  driven  in  the  same  direction  as  the  jet,  then 
vl  -f-  u  is  the  relative  velocity  of  the  jet  with  respect  to  the 
earth,  and  the  reaction  is 

R  =  horizontal  change  of  momentum 

w  w 

=  —Q(VI  +  u~)  =~<:svAvl(vl  +  ^ 

<~>  <5 

w 

=  j^»i(*i  +  «).      ........    (7) 

disregarding  the  contraction  and  putting  cv  =  I. 


SCOTCH    TURBINE. 


375 


9.  The  Jet  Reaction  Wheel  (Scotch  Turbine).— In  this 
fejni  of  motor  the  water  enters  the  centre  of  the  wheel, 
spreads  out  radially  in  tubular  passages,  and  issues  from 
openings  at  the  ends  tangentially  to  the  direction  of  rotation. 


FIG.  212. 


FIG.  214. 


FIG.  213. 


FIG.  215. 


Fig.  2 1 2  represents  the  simplest  wheel  of  this  class.  In 
England  it  is  known  as  Barker's  Mill,  and  in  Germany  as 
Segner's  Water-wheel. 

A  reaction  wheel  may  have  several  tubular  passages  as  in 
Fig.  214,  while  the  vertical  chamber  XY  may  be  cylindrical, 
prismatic,  or  conical. 

The  Scotch  -or  Whitelaw's  turbine,  Fig.  215,  does  not 
differ  essentially,  excepting  in  the  curved  arms,  from  the 
simple  reaction  wheel. 

Let  r  be  the  horizontal  distance  between  axis  of  orifice  and 

axis  of  rotation. 
"  h   lt     "    head  of  water  over  the  orifkes  when  closed, 


376  SCOTCH   TURBINE. 

Let  V  be  the  velocity  of  efflux  relatively  to  the  tube  when 

the  orifices  are  open. 
"   u   "     "    corresponding   linear    velocity  of  rotation   at 

the  centre  of  an  orifice. 

"  7'2  "     "    absolute  velocity  of  efflux  =   V —  u. 
"   Q  "     "    discharge. 
"   R  "     "    reaction. 
Then 

P.  =  */(«•  + V*),      .....      (I) 

cv  being  the  coefficient  of  discharge. 
Also, 

wQ 

(V  —  11)  —  horizontal  linear  change  of  momentum 

o 

=  reaction  producing  rotation 

=  R (2) 

The  useful  work 

=  Ru=  — r(r—u)u. (3) 


(4) 


The  efficiency 

Ru_    _  (V  —  u)u  _  2(V  ^  u)u 

~  wQh  ~         .gh  ~T~2         ~ ' 

-7~2  —  M 

cv 

Again,  the  efficiency 

_  (V  -  u)u  _  M*  IV         \ 
gh         ~  gh\u          I 

*  lfT    ,  ^-_ 
-  gh\l"         '    u*  I 

=  — j  j  ^Jl+  ~T  ""  terms  containing  higher  powers  of— j  —  I  [  . 

Thus  the  efficiency  must  theoretically  increase  with  u,  but 
the  value  of  n  is  limited  by  the  practical  consideration  that, 
even  at  moderately  high  speeds,  so  much  of  the  head  is 


SCOTCH   TURBINE.  377 

absorbed  by  frictional  resistance  as  to  sensibly  diminish  the 
efficiency. 

The  serious  defects  of  the  reaction  wheel  are  that  its  speed 
is  most  unstable  and  that  it  admits  of  no  efficient  system  of 
regulation  for  a  varying  supply  of  water. 

By  equation  (4),  the  efficiency  is  a  maximum,  for  a  given 
value  of  u,  when 

F2  -  2  Vu  +  cvV  =  o, 
or 


F=«(i  +  V    -<•/)  ......     (5) 

Experiment  also  indicates  that  the  best  effect  is  produced 
when  the  linear  speed  of  rotation  (u)  is  that  due  to  the  total 
head  (/z),  so  that 

y  =  2gh, 
and  therefore 

F2  =  vjgh. 

Substituting  these  values  in  equation  (5),  it  is  found  that 


_ 


and  hence,  by  equation  (4),  the  maximum  efficiency  =  f. 

Thus,  one  third  of  the  head,    i.e.,  —  ,  is  lost,   and  of  this 

7,2       (V-uf       h 

amount  the  portion  •-—  —  -  -   =  -,  is  carried  away  by  the 

2g  2g  9' 

effluent  water  in  its  energy  of  motion.      The  remainder,  viz., 

h       h        2  7 

—  —  =  —h  is  lost  in  frictional  resistance,  etc. 
399 

Ex.  A  reaction  wheel  with  six  tubular  passages,  each  of  4  sq.  ins. 
sectional  area,  passes  112,500  gallons  of  water  per  hour  and  makes  105 
revolutions  per  minute.  The  distance  between  the  axis  of  revolution 
and  the  axis  of  an  orifice  is  2  feet.  (Take  cv  =  i.) 

rr  4         i     112500          5 

F—1—  =  •?-?-.  —  /    ,    =  -?  cu.  ft.  per  sec.  per  orifice. 
144      66£.6o.  60       6 


378 


IMPACT   WHEEL. 


Therefore  V  —  30  ft.  per  sec. 

2  .  Tt  .  2.105 

Again,  u  = v =  22  ft.  per  sec. 

Hence,  if  h  is  the  head  over  the  orifices, 

302  =  222   +   2.32  .  //, 

and  h  =  6^  ft. 

624    5 
The  reaction  on  each  tube  =  — ^  .  ^(30  —  22)  =  13^  Ibs. 

The  useful  work  =  6  x  134*9  x  22  =  1718^  ft.-lbs. 

-  3i  H.P. 
ir 


The  efficiency  = 


13' 


io.  Impact  Wheel.  Borda  Turbine. — A  jet  moving  in 
the  direction  OC  (Fig.  216),  with  a  velocity  vl  (  —  OC)  im- 
pinges upon  a  flat  vane,  driving  it  in  the  direction  OE  with  a 
velocity  u  (=  OE).  Join  CE. 

9 


FIG.  216. 
Let  Ql  be  the  quantity  of  flow  towards  A,  and  ml  its  mass. 


Disregarding  the  effect  of  gravity,  which  is  equivalent  to 
the  assumption  that  the  movement  of  the  water  on  the  vane  is 
sensibly  in  a  horizontal  plane,  and  also  disregarding  friction, 
the  water  leaves  the  vane  at  A  and  B  with  a  relative  velocity 
V  =  Ag  —  Bg'  —  CE,  coincident  in  direction  with  AB  pro- 
duced. 


IMPACT   WHEEL  379 

Draw  A  A '  and  BB'  parallel  and  equal  to  OE  =  u. 
•f     Complete  the  parallelograms  A'g  and  B'g' .      Then 
Ah,  —  z/2',  represents  in  direction  and  magnitude  the  absolute 
velocity  with  which  the  water  leaves  the  vane  at 
A,  and 

B/i't  =  v.,",  represents  in  direction  and  magnitude  the  absolute 
velocity  with  which    the  water  leaves  the  vane 

at  A 

From  the  triangle  OCE, 

V*  =  vz  +  u*-  2Vlu  cos  y. 

From  the  triangle  AA'h, 

z/2' 2  =  Vs*  +  u2  —  2  Vu  sin  0. 

From  the  triangle  BB'h', 

v»t,  —   V2  +  u2  +  2Vu  sin  0. 
Hence 

— —   — —  =  u(vl  cos  y  —  u  +  V  sin  0) 

and 

»_«  —  »."•• 

=  u(vl  cos  y  —  u  —  V  sin  0). 


2 

Also, 


and 


w  QJ  cos  0\ 

=  J^-*SS7J 


w  22f          cos 

mn  =  -    — Vv,  — 


cos  0/' 

Therefore  the  useful  work 


=  !5!iu  _  ^m^ijg^  cos  v-«)  +  (a- GJ^sin 

,      /T-     ^7;     \      1  /-»rf~vr<     /J  /  \       1 

A    v\ 


38°  IMPACT   WHEEL. 

where 

0=0,+  Or 

If  the  directions  of  motion  of  the  vane  and  of  the  impinging 
jet  coincide, 

y  =  0  -\-  0  =  o     and      V  =  v^  —  u, 

and  therefore  the  useful  energy  imparted  to  the  vane 


w  u 


For  a  maximum  effect  vl  — 


Series  of  Vanes.  —  If  a  number  of  vanes  are  successively 
introduced  at  the  same  point  in  the  path  of  the  jet,  then 

w  w 

™i=~Qi     and     ™2=~Qr 

Thus  the  useful  energy  becomes 

IV 

-u\  Q(v,  cos  y-u)  +  (Ql  -  Q2)  V  sin  0}  ; 

<5 

and  if  the  directions  of  motion  of  the  vane  and  the  impinging 
jet  coincide, 

y  —  8  +  0  =  o,  v  V  —  vl  —  u, 

and  the  useful  energy 


w 


=  -u(v^  -  u)(Q  +Ql-Q2  sin  0). 

<5 

For  a  maximum  effect  vl  =  2u. 

Flow  in  One  Direction. — If  the  whole  of  the  water  flows 
away  in  the  direction  OA  so  that  <22  =  o  and  Ql  =  Qt  the 
useful  energy  for  a  single  vane 

wQ  u  I  cos  0\ 

=  ""      ~ u    ~ cos   ~u    Fsm 


IMPACT  WHEEL. 
and  the  useful  energy  for  a  series  of  vanes 


38r 


wO 

—  u(i'1  cos  y  —  u  + 

o 


sn 


For  a  given  value  of  0  this  last  is  greatest  when  y  (=  0  -\-  0} 
=  o,  and  therefore  V  =  v^  —  u.      Then 

wQ 
maximum  useful  energy  — u(y\  —  u)(l  4~  s^n  0)> 

<S 

which  increases  with  0  or  as  the  angle  of  exit  OAA' 
(—  90°  —  0)  diminishes,  indicating  that  it  is  advantageous  to 
curve  the  outlet  lip  of  the  vane. 

Denote  the  exit  angle  by  e,  Fig.  217.      Then 

J/2  —  U2  _[_  v*  —  2m\  cos  Y 
and 


Thus    the  useful  energy  imparted 
to  the  vane 

wQv 


—    -  —  u(?\  cos  y  —  u  -\-  V  cos  e). 

If  ^  is  so  small  that  cos  e  =  I,  ap- 
proximately, then  the  useful  energy 

=  - 
& 

wQ 

=  — 

<5 


cos  y  —  u  -}-  V] 


i~    cos       — 


-j-  \'uz  +  v?  —  2m\  cos  7). 
This  is  greatest  and  — when 


=  -J^  sec   y,  which 


is  the  best  speed  of  the  wheel.      In  such  case  the  whole  of  the 
jet's  energy  is  transformed  into  useful  work. 


BORDA'S    TURBINC. 


In  the  simplest  kind  of  impact  wheel  the  jet  strikes  the 
vane  more  or  less  perpendicularly  and  spreads  over  the  surface 
in  all  directions.  Wheels  of  5  ft.  diameter  are  used 
for  falls  of  from  10  to  20  ft.  The  vanes  are  I  5  ins. 
X  8  ins.  to  10  ins.  measured  radially,  and  are 
inclined  at  from  50°  to  70°  to  the  horizon.  The 
water  strikes  the  vane  in  a  direction  making  an 
angle  of  from  10°  to  20°  with  the  horizon,  i.e., 
nearly  at  right  angles. 
In  a  Borda  turbine  (Fig.  219)  revolving  about  a  vertical 
axis  OOj  the  vanes  are  curved  and  the  water,  as  it  flows  over 
them,  acts  principally  by  pressure.  The  vanes  are  set  between 
two  concentric  drums  which  should  be  of  considerable  depth 


v^ 

Q  -=; 

^=3 

. 

1 
1 

/ 

!    1 

/ 

i   i 
i   i 

/ 

i 

i  i 

i 

:  i 

i 

/ 

|   / 

' 

i  /    / 

i  /    / 

i  i    / 



•fi-/- 

_,.^ 

'"" 

u/  /  " 

^^ 

FIG.  219. 


FIG.  220. 


and  should  have  a  large  mean  diameter.  Borda  found  that 
with  such  a  turbine  an  efficiency  of  75  per  cent  could  be 
obtained  under  favorable  conditions.  As  the  water  passes 
through  the  turbine  the  fluid  particles  move  wholly  in  cylin- 
drical surfaces,  and  there  is  little  if  any  change  in  the  distance 
of  a  particle  from  the  axis.  Thus  the  effect  of  centrifugal 
force  may  be  disregarded. 

Let  Fig.  220  be  a  section  of  the  vane  at  a  distance  from 
the  axis  equal  to  the  mean  radius. 

The  water  strikes  the  vane  at  a  in  the  direction  ac,  falls 


BORON'S    TURBINE.  383 

upon  the  vane  through  a  vertical  distance  //2 ,  and  is  discharged 
atyin  the  direction///  with  a  velocity  7'.,. 

Let  F!  ,    F2  be  the  relative  velocities  ad  at  ^  and  ^-  at  /, 
respectively.      Then 

V}  =--  V?  +  2gh.1. 

If,  again,  the  angle  of  exit  e  at/is  so  small  that  cos  e  =  I, 
approximately, 


Suppose  that  the  water  leaves  the  turbine  without  energy, 
i.e.,  so  that  z>2  =  o  =  F9  —  n,  then 


=  u        v    --  2in\  cos  y 
and 

2/«'     COS         —   7/2  2// 


=  2g(H 


or 

UVt  COS  y  -.=  gHT  , 

an  equation  giving  the  best  speed  of  the  turbine. 

H  is  the  head  required  to  give  the  velocity  v^  at  entrance. 

Hl  is  the  total  head  under  which  the  turbine  works. 

There  should  be  no  loss  in  shock  at  entrance,  and  to  insure 
this  ad  (—  Fj),  the  relative  velocity,  must  be  tangential  to  the 
lip  at  a. 

The  lip  angle  a  is  then  given  by 

u        sin  (a  +  y) 
-  =  --  -  —        -  —  cos  y  4-  cot  a  sin  y, 

7     '  7 


3^4  BORON'S    TURBINE. 

or 

u 
cot  a  —  —  cosec  y  —  cot  y. 

v\ 

Since  u  =  V2,  the  triangle  fgh  is  isosceles,  and 

e 

v9  —  221*  sin  — . 
2  2         2 

wQ   i         v  *^\ 
The  useful  work  =  —  77(1 ), 

if  being  the  efficiency. 

Let  R  be  the  mean  radius. 
"  /    "      "    water  thickness,  measured  radially. 

Then, 

sin  c  =  Q. 


Allowance  may  be  made  for  the  principal  hydraulic  resist- 
ances (friction,  etc.)  by  taking 

f2  t-L  to  represent  the  loss  of  head  up  to  the  inlet,  and 

f. — ?-    "          '*          4<       "     <4     "      in  the  wheel-passages. 

*g 

Then 


and 


'2' 


/2  and/4  being  coefficients  to  be  determined  by  experiment. 

Usually /2  varies  from  .025  to  .2  and  upwards,  an  average 
value  being  .125,  and/4  varies  from  .1  to  .2. 

The  normal  distance,  Fig.  221,  between  two  consecutive 
vanes  should  be  >  the  stream's  normal  thickness  between  the 


BURDIN'S    WHEELS. 


385 


vanes,  i.e.,   >  v?  X  the  normal  thickness  of  the  stream  before 

m 

impact. 


FIG.  221. 


FIG.  222. 


Burdin's  (Fig.  222)  is  among  the  best  of  impact  wheels, 
differing  only  from  the  simple  Borda  in  receiving  the  water  at 
several  points  simultaneously  and  in  distributing  the  outlet 
openings  in  three  concentric  circles. 

Ex.  A  5-H.P.  Borda  turbine,  of  4  ft.  mean  diameter  and  4  ft.  depth, 
works  under  a  total  head  of  20  ft.  The  direction  of  the  jet  before  impact 
is  inclined  at  33°  33'  (y}  to  the  horizon,  and  the  angle  of  exit  (e)  is  19°  8'. 
The  jet  delivers  3  cu.  ft.  of  water  per  second.  Find  (a)  the  best  speed 
of  the  turbine;  (b)  the  lip  angle  a;  (c)  the  velocity,  2/a ,  of  the  water  as  it 
leaves  the  turbine;  (d)  the  hydraulic  efficiency;  (e)  the  practical  effi- 
ciency. 


20  —  4  =  16  —  head  required  to  produce  t/j  =  —  . 


(a} 
Therefore  v,.  =  32  ft.  per  second. 

The  best  speed  is  then  given  by 

uv\  cos  Y  =  £rffi> 
or  «  .  32  •  cos  33°  33'  =  32-20 


or 


u  .  32  =  32.20  .—     and     u  =  24  ft.  per  second. 


The  number  of  revolutions  per  minute  = 


' 


sin 


Z)==^.  =  fi==i  = 
2/1       32      4 


cos  y  -f  cot  ct  sin 


or 


and 


cot  (i  80°  -  a)  =  cot  33°  33'  —  —  cosec  33°  33'  =  .1509, 
a  =  98°  35'. 


386 


DANAIDES. 


(c)  Assuming  Fa  =  u,  then 

•z/a  =  2«  sin  —  =  48  sin  9°  34'  =  48  x  ^  =  8  ft.  per  second. 


(d)  The  hydraulic  efficiency  =  i  — 


=  ~  =  -9S- 


64. 20         20 

(e)  If  77  is  the  practical  efficiency, 

7.621.3(20-5-^)  =  5.550 

and  rj  =    .772. 

Danaides. — These  are  wheels  capable  of  revolving  about  a 
vertical  axis  and  consist  of  two  casings  which  are  more  or 
less  in  the  form  of  inverted  truncated  cones  (Fig.  223)  and 
which  enclose  a  space  divided  into  a  number  of  water-passages 
by  vanes  which  may  be  flat,  spiral,  or  screw-shaped.  In  the 
wheels  described  by  Belidor  the  inner  casing  with  the  vanes 


FIG.  223.  FIG.  224. 

attached  is  made  to  closely  fit  the  outer  conical  casing,  which 
is  fixed.  In  another  form  of  Danaide  the  vessel  is  divided 
into  two  equal  parts  by  a  vertical  partition.  Thus  in  wheels 
of  this  type  the  water  approaches  the  axis  in  its  descent, 
developing  a  centrifugal  force  which  must  be  taken  into 
account. 

Consider  the  case  of  a  Danaide  with  double  conical  casing 
and  flat  vertical  vanes,  Fig.  224. 


DANAIDES. 


3*7 


The  relative  velocities  Vl  ,   F2  are  evidently  at  right  angles 
to  ,  the  corresponding  peripheral  velocities  at  inlet   and  exit. 

A 

Therefore 

v*=   V?  +  u?     and     v*=  F22  +  u*. 

Also,  if  h2  is  the  depth  of  the  wheel, 

772        772  u  2_  uz 

V  2     _     *    1        I  _    ffl  _  ?!. 

" 


"  2,f  2g 

u      _ 

the  term     2  —       L  being  due  to  the  effect  of  centrifugal  force. 
<> 

Hence 


and  the  mechanical  effect 


Tzib-wheel.  —  This  form  of  impact  wheel,  Fig.  225,  consists 
of  a  number  of  floats  fixed  to  a  vertical  shaft.  The  wheel  is 
either  fitted  into  a  well,  a  small  clearance 
being  allowed,  or  it  is  given  a  larger 
diameter  and  is  placed  just  below  the 
well.  The  water  is  brought  along  a 
properly  designed  race,  enters  the  well 
tangentially  with  considerable  velocity 
and  acquires  a  rotary  motion.  Thus  it 
acts  upon  the  floats  both  by  impact  and 
by  pressure.  The  efficiency  of  the  wheel 
is  small,  as  a  large  portion  of  the  water 
escapes  without  producing  its  full  effect.  Practical  experience 
indicates  that  the  best  speed  of  the  middle  of  the  floats  is  about 
one  third  of  the  velocity  of  the  current,  and  that  the  efficiency 
varies  from  15  to  40  per  cent,  but  rarely  exceeds  30  per  cent. 


FIG.  225. 


IMPACT  ON   CURBED    l/ANES. 

ii.  Jet  impinging  upon   a  Curved  Vane  and   deviated 
wholly  in  one  Direction — Best  Form  of  Vane. — Let  the  jet, 

of  sectional  area  A,  moving  in  the 

T\  direction   AB    with    a    velocity    v^, 

v\    \          drive  the  vane  AD  in  the  direction 


AC  with  a  velocity  u,  Fig.  226. 

°\aj~~                  |  Vx   ^          Take  A  B  to  represent  v^  in  direc- 

\                      /  ^~p    tion  and  magnitude. 

\                 /  Take  AC  to  represent  u  in  direc- 

\           /  ^  tion  and  magnitude. 

\  J°in  CB- 

"V"-""" ~^B  Then    CB   evidently    represents 

^C  /  ^  ^e  velocity  of  the  water  relatively 

L  to  the  vane,  in  direction  and  magni- 

FIG.  226.  tude.      If  CB  is  parallel  to  the  tan- 

gent to  the  vane  at  A,  there  will  be  no  sudden  change  in  the 
direction  of  the  water  as  it  strikes  the  vane,  and,  disregarding 
friction,  the  water  will  flow  along  the  vane  from  A  to  D  with- 
out any  change  in  the  magnitude  of  the  relative  velocity 
V (=  CB).  The  vane  is  then  said  to  "  receive  the  water  with- 
out shock." 

Again,  from  the  triangle  ABC,  denoting  the  angles  BACY 
ABC,  ACB,  by  A,  B,  C,  respectively. 

u  __  A  C   _  sin  B  _          sin  B 
^  ~  ~AB  ~  sin  C  ~  sin  (A  +  B)J    '     '      ' 

and  therefore 

v 
cot  B  =  -  cosec  A  —  cot  A,    .      .      .     '.      (2) 

a  formula  giving  the  angle,  between  the  lip  and  the  direction 
of  the  impinging  jet,  which  will  insure  the  water  being  received 
"  without  shock." 

In   the  direction  of  the   tangent  to  the  vane   at  Z>,   take 
DE  =  CB  (=  V}. 


IMPACT  ON  CURBED   VAXES.  389 

Draw  DF  parallel  and  equal  to  AC  (=  u). 
,    Complete  the  parallelogram  EF. 

Then  the  diagonal  DG  evidently  represents  in  direction 
and  magnitude  the  absolute  velocity  ?>2  with  which  the  water 
leaves  the  vane. 

Draw  AK  equal  and  parallel  to  DG  (=  v2). 

Join  BK.  Then  BK  represents  the  total  change  of  velocity 
between  A  and  D  in  direction  and  magnitude. 

Thus  if  R  is  the  resultant  pressure  on  the  vane,  then 
R  =m.  BK. 

Let  ML  be  the  projection  of  BK  upon  AC. 

Then  ML  represents  the  total  change  of  velocity  in  the 
direction  of  the  vane's  motion. 

Let  P  be  the  pressure  upon  the  vane  in  this  direction. 

Then 

P  =  m  .  LM.      ......      (3) 

v  2  _  v  2 

The  useful  work  —  Pu  =  mil  .  LM  '=  m—      —  -.        .       4 


W       V 

The  total  available  work  =  —A-1-. 


g      2 


mu  .  LM  v?  —  v? 

The  efficiency  ^-^  =  ™*T^r  .....      (6) 

g       2 

Again,  join  CK. 

Then,  since  AC  is  equal  and  parallel  to  DF,  and  AK  to 
DG,  the  line  CK  is  equal  and  parallel  to  DE,  and  is  therefore 
equal  to  CB. 

Thus  in  the  isosceles  triangle  CBK,  CB  is  equal  and 
parallel  to  the  relative  velocity  V  at  A  ,  CK  is  equal  '  and 
parallel  to  the  relative  velocity  V  at  D,  and  the  base  BK  repre- 
sents the  total  change  of  motion. 


?9°  IMPACT  ON  CURBED   YANES. 

Let  d  be  the  angle  through  which  the  direction  of  the  water 
is  deviated,  i.e.,  the  angle  between  AB  and  AK.      Then 

F2=  CK^  —  AK*  +  AC*  —  2AK  .AC  cos  (A  +  6) 

—  V%  +  U2  —  2V2U  COS   (A  -h  <?),      ......        (7) 

and  also 


=  v*  +  u*  —  2.i\u  cos  A  .........      (8) 

Hence 

2  -     2 

*-  =  u\v^  cos  A  —  ?'?cos  (/4  4-  d)\.    .      .      (91 


If  BH  \s  drawn  parallel  to  the  tangent  at  D,  BK  evidenti> 
bisects  the  angle  between  BC  and  BH,  and  this  angle  is  equal 
to  the  angle  between  the  tangents  to  the  vane  at  A  and  D. 

Let  a  be  the  angle  between  the  normals  at  A  and  D. 
Then  the  angle  KCB  =  a,  and 

the  angle  CBK=  -(180°  -  a)  =  90°  -    -. 
Therefore 

BK  =  2<^(cos  90°  -  •?)  =  2  F  sin  ^. 
Hence 

R  =  m  .  BK  =  2mV  s'm  —  .        .      .      .      (10) 

Let  X,  Y  be  the  components  of  R  in  the  direction  of  the 
normal  at  A  and  at  right  angles  to  this  direction.  Then 

X  =  R  cos  -  —  ;«Fsin  a,    .......      (ll) 

ex.  a 

Y—  R  sin  —  —  2mVs'm2-  =  mV(\  —  cos  a).      .      (12) 
2  2 


IMPACT  ON    CURVED   VANES.  39  * 

The  efficiency  is  a  maximum  when 

d(Pu\  dP  , 

-A—  =  o  =  u-r-\-P.      ....     (13) 
du  du 

The  efficiency  is  nil  when 

Pu  =  o,      i.e.,  when  u  =  o  or  P  =  o.     .      .      (14) 

In  the  latter  case,  since  P  —  m  .  LM,   the  projection 
must  be  nil,  and  therefore  FIG.  227. 

BK    must    be     at    right 
angles  to  AC,  as  in  Fig. 

227. 


The  angle  ^4  (7.5  is  now 
=  1  80°  —  -.      Therefore 

u     _  sin  ABC 
v,  ~  sin  ACB 


or 

sin  - 

2 


If  ^7T  is    parallel    to 
",   Fig.    228,   then    the 
angle 


FIG.  228. 


and  therefore 


sn 


vl       sin  ACB 


COS 


392  IMPACT  ON  CURBED   YANES. 

Let  the  direction  of  the  impinging  jet  be  tangential  to  the 
vane  at  A,  Fig.  226,  and  let  the  jet  and  vane  move  in  the  same 
direction.  Then 


V  •=.  vl  —  u,     m  =  —  A(vl  —  u)  ; 

<5 

P—Y  —  —A(yl  —  u)\i  —  cos  a)  =  2~A(i\  —  «)2sin2-; 

<5  O 

useful  work  =  Pu  =  2  —  Au(vl  —  u)2  sin2  -; 


rr     . 

efficiency  —  4 


This  is  a  maximum  and  equal  to  —  sin2  —  when  i\  —  ^u. 

These  results  are  identical  with  those  for  a  concave  cup 
when  a  —  180°. 

Instead  of  one  vane  let  a  series  of  vanes  be  successively 
introduced  at  short  intervals  at  the  same  point  in  the  path  of 
the  jet.  Then 

w 
m=-.—Avl9 

and  hence  the  pressure  P,  useful  work,  and  efficiency  respec- 
tively become 

w  A  w        v?  —  v*       v*  —  v* 

—Av,.LM\      —Av±       -i;      -1 — ^—^. 
g  g  2  v? 

N.B.  Frictional  resistance  may  be  taken  into  account  by 
assuming  that  it  absorbs  a  fractional  portion  of  the  head  corre- 
sponding to  the  velocity  of  the  jet  relatively  to  the  surface  over 

F2 
which  it  spreads.      Thus  the   loss    of   head  =/ — ,   and  the 

o 

J/2 

corresponding  loss  of  energy  =  wQf — . 


EXAMPLE. 


393 


Ex.  A  curved  vane  in  the  form  of  the  quadrant  of  a  circle 
'without  shock,  at  an  edge,  a  stream  of  water  flowing  at  the  rate 
p&r  second,  which   drives  the   vane  with 
a  velocity  of  4  ft.  per  second  in  a  direc- 
tion making  an  angle  of  60°  with  the  re- 
ceiving edge. 

At  the  receiving  edge  the  triangle  abc  is 
a  triangle  of  velocities  in  which  the  angle 
abc  —  \20°,ac=  12  ft. ,#£  =  4  ft., and  be  =  V, 
the  relative  velocity  at  a  which  must  be 
parallel  to  the  tangent  at  at  as  there  is  to 
be  no  loss  in  shock.  Then 


receives 
of  12  ft. 


or 


and 


1 2'  =  4"  +  V 2  —  2.4  V  cos  1 20°, 
128  =  V  +  4^, 


V=  9.4891  ft.  per  second. 

Also,  if  y  is  the  angle  between  ac  and 
the  receiving  edge,  then  the  angle  cab  =  60°  —  y,   and 

9.4891  _  be  _  sin  (60°  —  y}        . 


or 


cot  y  =  3.3166     and  y  =  16°  47'. 


At  the  discharging  edge  fghk  is  the  parallelogram  of  velocities  in 
which  fg,  parallel  to  ab,  =  4  ft.,  fk,  tangental  at  /,  =  9.4891  ft.,  the 
relative  velocity,  and/^  is  the  absolute  velocity  in  direction  and  magni- 
tude with  which  the  water  leaves  the  vane.  Let  the  angle  hfk  =  S. 
Then 

TV  =  42  +  (9-489I)2  —  2  x  4  x  9.4891  cos  30°  =  40.2993, 
and  Vi  =  6.3481  ft.  per  sec. 

891       fk      sin  (d  +  30°) 


Again, 
and 


. 
=  cos  30"  +  s,n  30"  cot  5, 


cot  8  —  3.0126,     or     8  •=.  18°  22'. 


12.  Tangential  or  Centrifugal  Turbines.  —  Suppose  that 
the  vane  AD  is  constrained  to  revolve  about  a  vertical  axis  O 
with  a  constant  angular  velocity  GO.  If  OP,  OQ  are  consecu- 
tive radii  and  if  PN  is  drawn  at  right  angles  to  OQ,  then  the 


394  TANGENTIAL    TURBINES. 

work  of  the  centrifugal  force  as  a  mass  m  of  fluid  moves  from 
P  to  the  consecutive  position  Q 

=  ma?r  .  QN 
=  mo&r  .  dr, 
where  OP  =  r. 


FIG.  230. 
The  total  work  in  the  movement  from  A  to  D 

r* 

=     I     mo&r  dr 


i: 


Uj ,  U2  being  the  linear  velocities  at  A  and  D  respectively. 

u.a  — u2  . 

If  the  flow  is  from  A  to  D,  — —       -  is  evidently  a  gain  of 

2g 
head,  while  it  is  a  loss  of  head  if  the  flow  is  from  D  to  A. 

In  tangential  or  centrifugal  flow  turbines  a  number  of  vanes 
are  encased  and  have  concentric  inlet  and  outlet  surfaces. 
The  flow,  which  is  more  or  less  radial,  is  towards  the  axis  in 
the  inward-flow  and  from  the  axis  in  the  outward-flow  turbine. 
Since  the  axis  of  rotation  is  vertical,  the  effect  of  gravity  may 
be  disregarded. 


TANGENTIAL    TURBINES. 


395 


If  vr't  vr"  are  the  radial   components   of  vl  and  v2  respec- 
jtively, 

vr'  —  vl  sin  y     and     vr"  —  F2  sin  ft. 


FIG.  231. 


FIG.  232. 


Then,   by    the  condition    of  continuity  of  flow,   and    dis- 
regarding the  thickness  of  the  vanes, 


2nr^d^Ur  =  ^7^X1^1  sin  y  —  Q  = 
and 

Q 


r"  —  2nr.2d2V2  sin  ft, 


sin  A      ...      (i) 


and  ^2  being  the  inlet  and  outlet  depths  of  the  wheel. 
First.   Disregard  hydraulic  resistances.      Then 


y  2  _     Y  2     I  2  _        2 

r  2     ~         F   T     ~f-  «2  «j 


.       .       (2) 


cos 


-*• 


=  v*  —  2u^  cos  y  4.  ^22, 


or 


COS       . 


(3) 


396  TANGENTIAL    TURBINES. 

Also,  from  the  triangle 


*,'= 


-2  J>,  cos  /»  ......     (4) 


Hence 

wQ  v  2  —  v  2 
the  useful  work  =  —  —  -  -  -  - 

wQ 

=  —  ("ivi  cos  Y  +  «a  F2  cos  £  ~  «22)-         (5) 
<s 

The  energy  carried  away  by  the  water  on  leaving  the  tur- 
bine should,  of  course,  be  as  small  as  possible.  Two  assump- 
tions are  usually  made  in  practice,  viz.  , 

either  u2  =  V2  , 

and  then  also,  by  eq.  (2),  u^  —  Vl  ,  so  that  the  triangles  fkh 
and  acd  are  isosceles  ; 

or  d  —  90°, 

so  that  the  flow  at  outlet  is  radial,  or  ^2  =  vr"  ',  and  therefore 
the  tangential  component  of  v2  ,  or,  as  it  is  called,  the  outlet 
velocity  of  whirl,  is  nil. 

Adopting  the  assumption  uz=  F2,  in  which  case  the  tri- 
angles fkh  and  acd  are  isosceles,  then 


(6) 


Hence  eq.  (i)  becomes 


Q  r?  ,  v,  sin  fi 

—  =  r2d2u2  sin  ft  =  ~- 


or  rfd^  sin  2y  =  r*d2  sin  ft,       ....      (7) 

and,  by  eqs.  (5)  and  (6), 


the  useful  work  =  ™?    +  u}  cos  ft  - 


(r      2  \          r* 


TANGENTIAL    TURBINES.  397 

The  corresponding  efficiency 


•  ^ 

sin2  - 


(9) 


rx  cos  y 
Adopting  the  assumption  d  =  90°,  then 

vrr  cot  ft  —  v2  cot  /?  =  «2  =  F2  cos  ft.    .      .     (10) 
Eq.  (i)  now  becomes 

<2  ^2 

rldlvl  sin  ;/  =  -—  =  r2</2//2  tan  ft  —  —d^  tan  /?,      (i  i) 

and,  by  eqs.  (i),  (5),  and  (10), 

,       wQ  iuQvfrfd,  sin  2y 

the  useful  work  =  ---  u,v,  cos  y  =  -  (12} 

f'11  £2    rjd^  tan  ft 


The  corresponding  efficiency 

_  rX  sin  2y 


'   r*d2  tan  ft  ' 

Second.    The  principal  hydraulic  resistances  may  be  taken 

v  a 
into  account  by  taking  the   loss   of  head  up  to  inlet  —  /2-x  , 

o 

Vz 
and  the  loss  of  head  in  the  wheel-passages  =f4—2-tso  that  the 

<*> 

total  loss  due  to  the  resistances  in  question 


Eq.  (2)  now  becomes 

(l  +/4)^r        *i  +  U2  —  u\  >     •     •     •     (15) 
and  if  H  is  the  head  over  the  inlet, 

\  =  H.       ........     (16) 


39 8  EXAMPLES. 

Ex.  i.  A  centrifugal  inward  flow  turbine,  with  equal  inlet  and  outlet 
depths  and  working  under  the  head  of  200  ft.,  passes  i  cu.  ft.  of  water 
per  second.  The  angle  y  is  15°;  w\  =  $r?. ;  and  it  is  assumed  that 
u-i.  =  J/i.  Find  (a)  the  peripheral  speed;  (b)  the  lip  angle  at  outlet;  (c) 
the  energy  carried  away  by  the  water;  (d)  the  energy  lost  in  hydraulic 
resistance;  (e)  the  useful  work  ;  (/)  the  efficiency.  (Disregard  the  thick- 
ness of  the  vanes.) 

sin  15°  —  .259,     cos  15°  —  .966,     and  let    /*=/*  =  .125. 

/          i\z/ia 
(«)       *  +  Q]Z~  =  20°-      Therefore  z/i  =  io6f  ft.  per  sec. 


By  eq.  (14),  V?  +  u<?  -  us  =  |  VS  =-f  * 

or  Vi1  —  2z/i//i  cos  15°  =  -5-  #2*  =  -5-1  — 


.2 

sin 


or  2/1  =  1.972^!  =  io6f, 

so  that  //i  =  54.09  ft.  per.  sec. 

i  //,        sin(a+  15°) 

Also,  — -  =  - —  = : =  cos  15    +  cot  a.  sin  ] 

1.972       7/i  sin  a 

Therefore     cot  (180°  —  a)  =  cot  15°  — —  =  1.77277, 

1.972 

and  a=  145°  40". 

(b)  By  eq.  (i),          r\v\  sin  y  =  ra»a  sin  ft  =  - 

or  sin  ft  =  (~~\   x   1.972  x  .259  =  .79804, 

and  ft  =  52°  56'. 

(c)  The  energy  carried  away  by  the  water 

=  62^.  i .—  =  -^-4«a2  sin2  —  =  —  «!2(i  —  cos 
64        128  2        4 

=  -(54-09)2  x  -397  =  H5 '  -8  ft.-lbs. 
4 

=  2.64  H.P. 

(d)  By  eq.  (14),  the  loss  in  hydraulic  resistance 


=  2.94  H.P. 


EXAMPLES.  399 

(<?)  The  total  possible  work  =  62^  .  i  .  200  =  12,500  ft.-lbs. 

The  useful  work  =  12500  —  1451.8  —  1617.5  =  943°-7  ft.-lbs. 

=  17.146  H.P. 

(/)  The  efficiency  =  =  .754. 


Ex.  2.  A  centrifugal  outward-flow  turbine  with  an  efficiency  of  80 
per  cent  and  working  under  the  head  of  200  ft.  over  the  inlet  passes 
i  cu.  ft.  of  water  per  second.  The  angle  y  =  15°  ;  5ri  =  4^2 ;  and  the 
velocity  at  outlet  is  radial,  i.e.,  <5  =  90°.  Find  (a)  the  peripheral  speed  ; 
(b)  the  lip  angle  at  inlet ;  (c]  the  ratio  of  the  inlet  to  the  outlet  depth  ; 
(d)  the  lip  angle  at  outlet ;  (e)  the  energy  carried  away  by  the  water  ; 
( '/)  the  useful  work.  (Disregard  the  blade  thickness  and  the  hydraulic 
resistance.) 

(a)      TT-  =  200,     and  therefore     v±  =  804/2  —  113.17  ft.  per  sec. 

U\V\  COS    15°          Ui  COS   I  C° 

But  .8  —  the  efficiency  =  —  t-  = -—-, 

32  x  200  404/2 

or  Ui  =  324/2  sec  15°  —  46.851  ft.  per  sec. 

324/2"seci5°  _  2  _  «,  _  sin(a+  15°) 

it/)  , —  oCv*    1  K •    

804/2  5  vi  sin  a 

==  cos  15°  +  cot  a  sin  15°, 

and  cot  (180°  —  a)  =  cot  15°  —  —  sec  15°  cosec  15° 

—  3.7320508  —  1.6  =  2.1320508, 

and  «=  154°  52'- 

(<:)  By  eqs.  (10)  and  (12), 
u**(i  +  tan2  ft)  —  F22  ;=   Fi2  +  «2a  —  u<?  =  u<?  +  v?  —  iu\v\  cos  Yt 

or  «2a  tan3  /S  =  f  — J   f^-J  v?  sin2  15°  =  z/i2  —  2u^i  cos  15, 

or  w22tan2/?  =  8192  sin2  15°  f-^-J  =  804/2(804/2"—  644/2^  =  2560, 

which  gives  f^-J  =  4.665, 

.and  therefore  -j-  =  2.16. 

(d)  By  (c\ 

«,-  m-  ,r  -^-  COS2  1 5°  =  .7464- 


400  JET   TURBINE. 

Therefore  tan  ft  =  .864, 

and  ft  —  40°  50'. 

0)  The  energy  carried  away  by  the  water 


=  2500  ft.-lbs. 
=  4T«TH.P. 

(/)  The  total  possible  work 

=    1.62^.  200 

=  12,500  ft.-lbs. 
Hence  the  useful  work 

=  total  possible  work  —  2500 
—  12500  —  2500  =  10,000  ft.-lbs. 
=  i8T"TH.P. 

13.  Jet  Turbine.  —  In  the  jet  turbine  the  water  passes  along 
the  axis  and  is  distributed  radially  in  all  directions  so  that  the 
angle  y  =  90°.  It  is  no  longer  possible  to  have  ul  =  Vl  ,  and 
it  cannot  therefore  be  assumed  that  u2  =  Vr  A  fair  efficiency 
may,  however,  be  secured  by  making  u2  =  vr 


FIG.  233. 

First,   disregard    hydraulic    resistances.      Then,   from    the 
triangle  adc, 

u*  +  v*  =  V?  =  V}  -  u?  +  u? 
=  V?  -  v?  +  u?. 


JET   TURBINE.  40* 

and  2v?  =   F22. 

^rom  the  triangle//^, 

z,22  =  u*  +  J722  -  2«2F2  cos  /? 

=   ^2(3   -2   4/2  COS  /ff). 

Hence 

wQ  v*  -  v* 
the  useful  work  —   - 

S          2 

7'2 

=  ze/g  —  (2  V2  cos  ft  —  2), 
and  the  efficiency  =  2  \  2  cos  ft  —  2. 

Hence,  too,  cos  /?>-—,  i.e.,  /?  must  not  exceed  45°. 

\2 

Second,  taking  the  hydraulic  resistances  into  account, 


V  +  v?  ='-   17  =  ('  +/<)  V}  -  u?  +  u 


Also,  the  loss  of  head  up  to  inlet  =f2-±-m 

V*         f 
"      "     "    "  in  wheel-passages  =/4—^  = 


and  the  total  loss  of  head  due  to  the  principal  hydraulic  resist- 
ances 


If  //  is  the  head  over  the  inlet, 

o  +/*)--  =  H. 


NOTE. — Impact,   centrifugal,   and   jet    turbines  will    work 
with  the  axis  inclined  at  any  angle  to  the  vertical. 


402  RESISTANCE    TO  MOTION  OF  SOLIDS  IN  FLUIDS. 

14.  Resistance  to  the  Motion  of  Solids  in  a  Fluid  Mass. 

—The  preceding  results  indicate  that  the  pressure  due  to  the 
impact  of  a  jet  upon  a  surface  may  be  expressed  in  the  form 

F2 

P=  KwA—, 
ig 

A  being  the  sectional  area  of  the  jet,  V  the  velocity  of  the  jet 
relatively  to  the  surface,  and  K  a  coefficient  depending  on  the 
position  and  form  of  the  surface. 

Again,  the  normal  pressure  (^V)  on  each  side  of  a  thin 
plate,  completely  submerged  in  an  indefinitely  large  mass  of 
still  water,  is  the  same.  If  the  plate  is  made  to  move  hori- 
zontally with  a  velocity  F,  a  forward  momentum  is  developed, 
in  the  water  immediately  in  front  of  the  plate,  while  the  plate 
tends  to  leave  behind  the  water  at  the  back.  A  portion  of  the 
water  carried  on  by  the  plate  escapes  laterally  at  the  edges 
and  is  absorbed  in  the  neighboring  mass,  while  the  region  it 
originally  occupied  is  filled  up  with  other  particles  of  water. 
Thus  the  normal  pressure  Ar,  in  front  of  the  plate,  is  increased 
by  an  amount  ;/,  while  at  the  back  eddies  and  vortices  are 
produced,  and  the  normal  pressure  N  at  the  back  is  diminished 
by  an  amount  ;/'.  The  total  resultant  normal  pressure,  or  the 
normal  resistance  to  motion,  is  n  -f-  «',  and  this  increases  with 
the  speed.  In  fact,  as  the  speed  increases,  n  approximates 
more  and  more  closely  to  N,  and  in  the  limit  the  pressure  at 
the  back  would  be  nil,  so  that  a  vacuum  might  be  maintained. 

Confining  the  attention  to  a  plate  moving  in  a  direction 
normal  to  its  surface,  the  resistance  is  of  the  same  character 
as  if  the  plate  is  imagined  to  be  at  rest  and  the  fluid  moving 
in  the  opposite  direction  with  a  velocity  V.  So,  if  both  the 
water  and  the  plate  are  in  motion,  imagine  that  a  velocity 
equal  and  opposite  to  that  of  the  water  is  impressed  upon  every 
particle  of  the  plate  and  of  the  water.  The  resistance  is  then 
•of  the  same  character  as  that  of  a  plate  moving  in  still  water, 
the  velocity  of  the  plate  being  ,the  velocity^relatively  to  the 


RESISTANCE   TO  MOTION  OF  SOLIDS  IN  FLUIDS.  4°3 

Water.  Thus,  in  general,  the  resistance  to  the  motion  of  such 
a  -plane  moving  in  the  direction  of  the  normal  to  its  surface, 
with  a  velocity  V  relatively  to  the  water,  may  be  expressed  in 
the  form 

F2 
R  =  KwA  —  , 


A  being  the  area  of  the  plate,  and  K  a  coefficient  depending 
upon  the  form  of  the  plate  and  also  upon  the  relative  sectional 
areas  of  the  plate  and  of  the  water  in  which  it  is  submerged. 

According  to  the  experiments  of  Dubuat,  Morin,  Piobert, 
Didion,  Mariotte,  and  Thibault,  the  value  of  K  may  be  taken 
at  1.3  for  a  plate  moving  in  still  water,  and  at  1.8  for  a  current 
moving  on  a  fixed  plate.  Unwin  points  out  the  unlikelihood 
of  such  a  difference  between  the  two  values,  and  suggests  that 
it  might  possibly  be  due  to  errors  of  measurement. 

Again,  reasoning  from  analogy,  the  resistance  to  the 
motion  of  a  solid  body  in  a  mass  of  water,  whether  the  body 
is  v.'holly  or  only  partially  immersed,  has  been  expressed  by 
the  formula 

F2 

R  —  KwA  —  , 
V 

V  being  the  relative  velocity  of  the  body  and  water,  A  the 
greatest  sectional  area  of  the  immersed  portion  of  the  body  at 
right  angles  to  the  direction  of  motion,  and  K  a  coefficient 
depending  upon  the  form  of  the  body,  its  position,  the  relative 
sectional  areas  of  the  body  and  of  the  mass  of  water  in  which 
it  is  immersed,  and  also  upon  the  surface  wave-motion. 
The  following  values  have  been  given  for  K\ 

K  —  i.i  for  a  prism  with  plane  ends  and  a  length  from  three 
to  six  times  the  least  transverse  dimension  ; 

K  —  i  .o  for  a  prism,  plane  in  front,  but  tapering  towards  the 
stern,  the  curvature  of  the  surface  changing  gradually 


404  PRESSURE   ON  PLATE  IN  PIPE. 

so  that  the  stream-lines  can  flow  past  without  any  pro- 
duction of  eddy  motion,  etc.  ; 

JC  =.  .5  for  a  prism  with  tapering  stern  and  a  cut-water  or 
semicircular  pro\v; 

K  —  .33  for  a  prism  with  a  tapering  stern  and  a  prow  with  a 
plane  front  inclined  at  30°  to  the  horizon ; 

!£'  =    .16  for  a  well-formed  ship. 

Froude's  experiments,  however,  show  that  the  resistance 
to  the  motion  of  a  ship,  or  of  a  body  tapering  in  front  and  in 
the  rear,  so  that  there  is  no  abrupt  change  of  curvature  leading 
to  the  production  of  an  eddy  motion,  is  almost  entirely  due  to 
skin-friction  (see  Art.  i,  Chap.  II). 

15.  Pressure   of  a   Steady  Stream  in   a   Uniform  Pipe 

against  a  Thin  "Plate  AB  Normal  to  the  Direction  of  Motion. 

—The  stream-lines  in  front  of  the  plate  are  deviated  and  a 

contraction   is  formed  at  Cfy      They  then  converge,  leaving 

a  mass  of  eddies  behind  the  plate. 

Consider  the  mass  bounded  by  the  transverse  planes  ClCl> 
Cfz^  where  the  stream-lines  are  again  parallel. 

At   ClCl  let  /lf   Aiy  7'j,   2l  be  the  mean  intensity  of  the 
ct       r  pressure,   the  sectional  area  of  the 

waterway,  the  velocity  of  flow,  and 
the  elevation  of  the  C.  of  G.  of  the 
section  above  datum. 

Let   /2,    A2,    7'2,    22   be   corre- 
sponding symbols  at  Cf. 

Let /3 ,  Alt  7'1?  #3  be  corresponding  symbols  at  Cfy 
Let  a  be  the  area  of  the  plate. 
Let  cc  be  the  coefficient  of  contraction. 

Neglect  the  skin  and  fluid  friction  between  ClCl  and  Cfy 
Then,  by  Bernoulli's  theorem, 

.  A  ,  'i  =  =  8    ,  A  ,  !¥*  = :  g    ,  A  ,  *>?  ,  (v,  -  ^'.)3 

W          2g          *'2         W          2g          "*          U'         2g  2g         * 


PRESSURE  ON  PLATE  IN  PIPE.  405 

(TV  —  7;  )2 
the  term   -'»        *'-    representing  the  loss  of  -head  due  to  the. 

bending  of  the  stream-lines  between  C2C.t  and  Cf  . 
Hence 


Again,  let  R  be  the  total  pressure  on  the  plane.      Then 

,  -  p^A,  =  (A  -  /3)A  =  jfluid   Pressure  «n  the  direction 

(      of  the  axis, 


Thus 


=  component  of  the   weight  in  the  direction 
of  the  axis. 


i  +  wAfa  —  ^3)  —  R  —  change  of  motion  in  direc- 

tion of  axis 


=  o, 


since  the  motion  is  steady. 
Hence 

R      _  A  ~  A 


But  ^^  =3  ^2?,2  ^'^(^  -  «)z/2.      Therefore 


<  — 

(ce(m  —  i) 


406  PRESSURE  ON  CYLINDER  IN  PIPE. 

where  m  —  —  1,  or 

a 


R  = 


where  K  —  m  \—  --  r  —  I  !•  . 
\cc(m  -  i)          J 

16.  Pressure  of  a  Steady  Stream  in  a  Uniform  Pipe  on  a 
Cylindrical  Body  about  Three  Diameters  in  Length.—  The 
stream-lines  in  front  of  the  body  are  deviated  and  a  contraction 
is  formed  at  C2C9.  They  then  converge,  flow  in  parallel  lines, 
and  converge  a  second  time  at  C.£.6,  leaving  a  mass  of  eddies 
behind  the  body. 

Consider  the  mass  bounded  by  the  planes  ClCl  ,  C  \C  \. 
As  in  the  previous  article,  let 

pl  ,  Alt  "c\  ,  s^  be  the  intensity  of  pressure,  sectional  area 
of  the  waterway,  velocity  of  flow,  and 
elevation  of  C.  of  G.  above  datum  at 
G  C,  Cfr 

p,,  A,,  7-      ^r,  be  similar  symbols 

jr  2,  '  i  *        z  J 

•     for  C\CZ. 

°*       /3  ,  ^43  ,  ^3  ,  -s-3  be  similar  symbols 
FIG.  235.  for  Cfv 

Pi  ,  Alt  7't  ,  ^4  be  similar  symbols  for  C"46*4. 
Neglect  the  skin  and  fluid  friction  between  ClCl  and  C  \C4. 
Then,  by  Bernouilli's  theorem, 

Z* 


—  ^-  being  the  loss  of  head  between  CjC,  and  C3^3  and 

o 

/        _     ,  \2 

being  the  loss  of  head  between  Cfz  and  ^4£"4. 


•O 


PRESSURE  ON  CYLINDER  IN  PIPE.  4°7 

Hence 


, 

&  \       ~~~      &  4     ~\ 


A  -  A  _  fe  -  * 


w 

But  Alvl  =  A2vz  =  A^  ,     cc(Al-a)  =A2, 

"~  '  ' 
and  As  =  Al  —  a. 

•  ..    ;  "     '  •  '. 

Therefore 


m         -\ 

-i)  ~  ^P"i  j  j 


where  m  =  — L. 

Also,  as  in  the  preceding  article, 

Hence 


=  ^ 

K  =  m 


where  m  —  — L,  and 
i 


\(m~-  i)^  (m—  i 

This  value  of  K  is  always  less  than  the  value  of  K  for  the 
plate  in  the  preceding  article  for  the  same  values  of  m,  a, 
and  cc. 

Hence  the  pressure  on  the  cylinder  is  also  less  than  the 
corresponding  pressure  on  the  plate. 

In  every  qase  K  should  be  determined  by  experiment. 


4°8  EXAMPLES. 


EXAMPLES. 

1.  A  stream  with  ^  transverse  section  of  24  sq.  ins.  delivers  10  cu. 
ft.  of  water  per  second  against  a  flat  vane  in  a  normal  direction.     Find 
the  pressure  on  the  vane.  Ans.  1171^  Ibs. 

2.  If  the  vane  in  example  i  moves  in  the  same  direction  as  the  im- 
pinging jet  with  a  velocity  of  24  ft.  per  second,  find  (a)  the  pressure  on 
the  vane  ;  (b}  the  useful  work  done;  (c)  the  efficiency. 

Ans.  (a)  42 1  £  Ibs.;  (b)  10,125  ft.-lbs. ;  (c]  .288. 

3.  What  must  be  the  speed  of  the  vane  in  example  2,  so  that  the 
efficiency  of  the  arrangement  may  be  a  maximum  ?    Find  the  maximum 
efficiency.  Ans.  20  ft.  per  sec.;  /T. 

4.  Find  (a)  the  pressure,  (b)  the  useful  work  done,  (c}  the  efficiency, 
when,  instead  of   the  single  vane  in  example    2,  a  series  of  vanes  are 
introduced  at  the  same  point  in  the  path  of  the  jet  at  short  intervals. 

Ans.  (a)  703i  Ibs.;  (b}  16,875  ft.-lbs.  ;  (c)  .48. 

What  must  be  the  speed  of  the  vane  to  give  a  maximum  efficiency  ? 
What  will  be  the  maximum  efficiency?  Ans.  30  ft.  per  sec.;  .5. 

5.  A  stream  of  water  delivers  7500  gallons  per  minute  at  a  velocity  of 
15  ft.  per  second  and  strikes  an  indefinite  plane.     Find  the  normal  pres- 
sure on  the  vane  when  the  stream  strikes  the  vane  (a)  normally;  (b)  at 
an  angle  of  60°  to  the  normal.  Ans.  (a)  585  9  Ibs.;  292.9  Ibs. 

6.  A  railway  truck,  full  of  water,  moving  at  the  rate  of  10  miles  an 
hour,  is  retarded  by  a  jet  flowing  freely  from  an  orifice  2  ins.  square  in 
the  front,  2  ft.  below  the  surface.     Find  the  retarding  force. 

Ans.  7.97  Ibs. 

7.  A  jet  of  water  of  48  sq.  ins.  sectional  area  delivers  100  gallons  per 
second  against  an  indefinite  plane  inclined  at  30°  to  the  direction  of  the 
jet ;  find  the  total  pressure  on  the  plane,  neglecting  friction.     How  will 
the  result  be  affected  by  friction  ?  Ans.  750  Ibs. 

8.  If  the  plane  in  example  7  move  at  the  rate  of  24  ft.  per  second  in 
a  direction  inclined  at  60°  to  the  normal  to  the  plane,  find  the  useful 
work  done  and  the  efficiency.  Ans.  2250  ft.-lbs.;  ^. 

At  what  angle  should  the  jet  strike  the  plane  so  that  the  efficiency 
might  be  a  maximum  ?     Find  the  maximum  efficiency. 

Ans.  sin"1  f  ;  \. 

9.  A  stream  of  32  sq.  ins.  sectional  area  delivers  7^  cu.  ft.  of  water 
per  second.     At  short  intervals  a  series  of  flat  vanes  are  introduced  at 
the  same  point  in  the  path  of  the  stream.     At  the  instant  of  impact  the 
direction  of  the  jet  is  at  right  angles  to  the  vane,  and  the  vane  itself 
moves  in  a  direction  inclined  at  45°  to  the  normal  to  the  vane.     Find 


EXAMPLES.  409 

the  speed  of  the  vane  which  will  make  the  efficiency  a  maximum.     Also 
find  the  maximum  efficiency  and  the  useful  work  done. 

Ans.  15.08  ft.  per.  sec.;  -/r ;  2io6||f  ft.-lbs. 

10.  A  stream  of  water  of  £  sq.  ft.  sectional  area  delivers  16  cu.  ft.  per 
second  normally  against  a  flat  vane.     Find  the  pressure  on  the  vane. 

If  the  vane  moves  in  the  same  direction  as  the  impinging  jet,  with  a 
velocity  of  32  ft.  per  second,  find  (a)  the  pressure  on  the  vane;  (ti)  the 
useful  work  done  ;  (c]  the  efficiency. 

How  would  the  results  be  affected  if  the  vane  were  inclined  at  45°  to 
the  jet,  and  moved  in  the  direction  of  its  normal  with  a  velocity  of 
24  ft.  per  second  ? 

Ans.  4000  Ibs.;  2250  Ibs.,  72,000  ft.-lbs.,  ^;  1802.8  Ibs.,  43,268 
ft.-lbs.,  .169. 

1 1.  Two  cubic  feet  of  water  are  discharged  per  second  under  a  pres- 
sure of  100  Ibs.  per  sq.  in.  through  a  thin-lipped  orifice  in  the  vertical 
side  of  a  vessel,  and  strike  against  a  vertical  plate.     Find  the  pressure 
on  the  plate  and  the  reaction  on  the  vessel.  Ans.  475.82  Ibs. 

12.  A  stream  moving  with  a  velocity  of  16  ft.  per  second  in  the  di- 
rection ABC  strikes  obliquely  against  a  flat  vane  and  drives  it  with  a 
velocity  of  8  ft.  per  second  in  the  direction  BD,  the  angle  CBD  being 
30°.     Find  (a)  the  angle  between  ABC  and  the  normal  to  the  plane  for 
which  the  efficiency  is  a  maximum;  (b)  the  maximum  efficiency  ;  (c)  the 
velocity  with  which  the  water  leaves  the  vane;  (d)  the  useful  work  per 
cubic  foot  of  water. 

Ans.  (a]  21°  44';  (d)  .25664  ;  (c)  12.6  ft.  per  sec.;  (d}  256.64  ft.-lbs. 

13.  At  8  knots  an  hour  the  resistance  of  the  Water-witch  was  5500 
Ibs.;  the  two  orifices  of  her  jet   propeller  were  each   18  ins.  by  24  ins. 
Find  (a)  the  velocity  of  efflux  ;  (b)  the  delivery  of  the  centrifugal  pump; 
(c)  the  useful  work   done;  {d}  the  efficiency;  (e)  the  propelling  H.P.» 
assuming  the  efficiency  of  the  pump  and  engine  to  be  .4. 

Ans.  (a)  29.4  ft.  per  sec.;  (b}  1104.6  gallons  per  sec.;  (c)  74,393 
ft.-lbs.;  (d)  .63;  (e)  536.7. 

14.  If    feathering-paddles  are    substituted    for   the   jet    propeller    in 
question  15,  what  would  be  the  area  of  stream  driven  back  for  a  slip  of 
25$?     Find  the  efficiency  and  the  water  acted  on  in  gallons  per  minute. 

Ans.  34.63  sq.  ft.;  .75  ;  234.206. 

15.  A  jet  issues  horizontally,  under  a  head  of  20  ft.,  from  a  |-in.  ori- 
fice in  the  vertical  face  of  a  tank  and  strikes  normally  the  centre  of  a 
vane  at  a  distance  of  48  ins.,  measured  horizontally,  from  the  tank's  face. 
By  measurement  the  vertical  distance  of  the  point  of  impact,  below  the 
axis  of  the  orifice,  was  found  to  be  2.582  inches.     Find  the  coefficient  of 
velocity  (cv\  the  inclination  of  the  vane's  axis  to  the  horizontal,  and  the 
coefficient  of  impact  (a)  in  the  following  cases  : 

(a)  A  flat  i2-in.  circular  vane,  the  balancing  weight  (W)  being 
3.015  Ibs. 


410  EXAMPLES. 

(b)  A  hemispherical  vane  of  12  ins.  diameter,  W  being  3.556  Ibs. 
(<:)  A  hemispherical  vane  of  3  ins.  diameter,  W  being  5.776  Ibs. 

(d)  A  parabolic  vane  with  a  base  of  12  ins.  in  diameter  and  6  ins.  in 
height,  W7  being  3.535  Ibs. 

(e)  An  elliptic  vane,  6  ins.   in  height  and  having  a  base  of   12   ins. 
diameter,  W  being  3.56  Ibs.     The  vane  edge  is  inclined  at  20°  to  the 
axis. 

Ans.   .96411;  6°  8';  (a)  .9834;  (b)  -5799;   (c)  -94*9',   (ft)   .6086; 
(e)  .5986. 

16.  Find  the  angle  of   blade  at  entrance,  the  useful  effect,  and  the 
efficiency  of  a  Borda  turbine  from  the  following  data  :  the  jet  at  entrance 
makes  an  angle  of  60°  with  the  horizontal  ;  the  depth  of  the  turbine  is 
3  ft.;  the  total  fall  to  the  point  of  discharge  is  19  ft.;  the  mean  diameter 
of  the  turbine  is  4  ft.;  the  quantity  of  water  passing  through  the  tur- 
bine is  4cu.  ft.  per  second  ;  the  angle  of  blade  at  exit  is  30°.    (Disregard 
hydraulic  resistance.)  Ans.  51°  33';  7.256  H.P.;  .84. 

17.  What  advantages  are  gained  by  increasing  the  depth  and  diam- 
eter of  a  Borda  turbine  and  by  curving  the  outlet  lips  of  the  buckets  ? 

18.  A  Borda  turbine  of  3.5  ft.  mean  diameter  has  a  head  of  12.96  ft. 
over  the  inlet,  a  practical  efficiency  of  .75,  a  theoretic  efficiency  (i.e.,  dis- 
regarding hydraulic  resistances)  of  .9265  and  delivers  3  horse-power.    The 
radial  width  of  the  water-passages  is  3  ins.,  and  the  depth  of  the  turbine 
is  2.04  ft.     If  there  is  to  be  no  shock  at  entrance,  find  (a)  the  inlet  and 
outlet  lip  angles  ;  (b)  the  velocity  (v9)  of  discharge  ;  (<•:)  the  quantity  of 
water  used  by  the  turbine. 

Ans.    (a}  ill0  25',  25°  12'  ;  (b}  8.4  ft.  per  sec.;  (c)    2.532  cu.   ft. 
per  sec. 

19.  In  a  railway  truck,  full  of  water,  an  opening  2  ins.  in  diameter  is 
made   in   one  of  the  ends  of  the  truck,  9  ft.  below  the  surface  of  the 
water.     Find  the  reaction  (d)  when  the  truck  is  standing;  (b)  when  the 
truck  is  moving  at  the  rate  of  10  ft.  per  second  in  the  same  direction  as 
the  jet;  (c)  when  the  truck  is  moving  at  the  rate  of  10  ft.  per  second  in 
a  direction  opposite  to  that  of  the  jet.     If  this  movement  of  the  truck 
is  produced  by  the  reaction  of  the  jet,  find  the  efficiency. 

Ans.    (a]    24.55    lbs-  Per    sq-  '"•?    (^)    34-?8  Ibs.   per  sq.    in.; 
(c)    14.3  Ibs.  per  sq.  in.;  .588. 

20.  From  a  ship  moving  forward  at  6  miles  an  hour  a  jet  of  water  is 
sent  astern  with  a  velocity  relative  to  the  ship  of  30  ft.  per  second  from 
a  nozzle  having  an  area  of  16  sq.  ins.;  find  the  propelling  force  and  the 
efficiency  of  the  jet  as  a  propeller  without  reference  to  the  manner  in 
which  the  supply  of  water  may  be  obtained.        Ans.  138^  ibs.;  .4535. 

21.  A  reaction  wheel  is  inverted  and  worked  as  a  pump.     Find  the 
speed  of  maximum  efficiency  and  the  maximum  efficiency,  the  coeffi- 
cient of  hydraulic  resistance  referred  to  the  orifices  being  .125. 

Ans.  Speed  =  twice  that  due  to  lift;  .758. 


EXAMPLES. 

22.  A  reaction  wheel  with  orifices  2  ins.  in  diameter  makes  80  revolu- 
tions per  minute  under  a  head  of  5  ft.    The  distance  between  the  centre 
of  an  orifice  and  the  axis  of  rotation  is  12  inches.     Find  the  H.P.  and 
the  efficiency.  Ans.  .146  ;  .596. 

23.  In  a  reaction  wheel  the  speed  of  maximum  efficiency  is  that  due 
to  the  head.      In  what  ratio  must  the  resistance  be  diminished  to  work 
at  f  this  speed,  and  what  will  then  be  the  efficiency  ?     Obtain  similar 
results   when   the  speed  is  diminished  to  three  fourths  of  its  original 
amount.  Ans.  .94;  .8896;   1.067;  .75 

24.  In  a  reaction  wheel,  determine  the  per  cent  of  available  effect  lost 
•(i)  if  V*  =  2gH\  (2)  if  V1  =  4gH;  (3)  if  F2  =  ZgH. 

What  conclusion  may  be  drawn  from  the  results? 
Efficiencies  are  respectively  .828,  .9,  .945. 

25.  A  stream  of  64  sq.  ins.  section  strikes  with  a  4o-ft.  velocity  against 
-a  fixed  cone  having  an  angle  of  convergence  =  100°;  find  the  hydraulic 
pressure.  Ans.  492.1  Ibs. 

26.  A  jet  of  9  sq.  ins.  sectional  area,  moving  at  the  rate  of  48  ft.  per 
second,  impinges  upon  the  convex  surface  of  a  paraboloid  in  the  direc- 
tion of  the  axis  and  drives  it  in  the  same  direction  at  the  rate  of  16  ft. 
per  second.      Find  the  force  in  the  direction  of  motion,  the  useful  work 
done,  and  the  efficiency.     The  base  of  the  paraboloid  is  2  ft.  in  diameter 
and  its  length  is  8  ins.  Ans.   25  Ibs.;  400  ft.-lbs. ;  T|j. 

27.  A  stream  of  water  of  16  sq.  ins.  sectional  area  delivers  12  cu.  ft.  of 
water  per  second  against  a  vane  in  the  form  of  a  surface  of  revolution, 
snd  drives  in  the  same  direction,  which  is  that  of  the  axis  of  the  vane. 
The  water  is  turned  through  an  angle  of  60°  from  its  original  direction 
before  it  leaves  the  vane.     Neglecting  friction,  find  the  speed  of  vane 
which  will  give  a   maximum    effect.     Also  find  impulse    on  vane,    the 
work  on  vane,  and  the  velocity  with  which  the  water  leaves  the  vane. 

Ans.  36  ft.  per  sec.  ;  562^  Ibs.;  20.250  ft.-lbs.  ;  95.24  ft.  per.  sec. 

28.  A  surface  of  revolution  is  driven  in  the  direction  of  its  axis  with 
a  velocity  of  16  ft.   per  second  by  means  of  a  jet  of  water  of  18  sq.  ins. 
sectional  area,  which  moves  in  the  direction  of  the  axis  with  a  velocity 
of  80  ft.  per  second,  and  impinges  upon  the  convex  side  of  the  surface. 
The  tangent  at  the  edge  of  the  surface  makes  an  angle  of  30°  with  the 
vertical.     Find  the  pressure  on  the  surface  and  the  efficiency. 

Ans.   500  Ibs. ;  .128. 

29.  A  jet  of  water  under  a  head   of   20  ft.,  issuing  from  a  vertical 
thin-lipped  orifice  i  in.  in  diameter,  impinges  upon  the  centre  of  a  vane 
}  ft.  from  the  orifice.     Determine  the  position  of  the  vane  and  the  force 
of  the  impact  (a)  when  the  vane  is  a  plane  surface ;  (b)  when  the  vane  is 
6  ins.  in   diameter  and  in  the  form  of  a  portion  of  a  sphere  of  6  ins. 
radius. 

Ans.  (a)  13,679  Ibs.  ;  (&)  20,518  Ibs.  or  6. 839  Ibs.  according  as 
vane  is  concave  or  convex. 


412  EXAMPLES. 

30.  A   stream   of  water    i    in.   thick  and  8  ins.  wide,  moving  with  a 
velocity  of  18  ft.  per  second,  strikes  without  shock  a  circular  vane,  of  a 
length  subtending  an  angle  of  90°  at  the  centre.     The  vane  is  driven  in 
the  direction  of  the  stream  with  a  velocity  of  6  ft.  per  second.     Find 
the  pressure  on  the  vane,  the  work  done,  and  the  efficiency. 

Ans.  22/-g  Ibs.  ;  93!  ft.-lbs.  ;  JT. 

31.  A  Pelton  wheel  of  2  ft.  diameter  makes  822  revolutions  per  min- 
ute under  a  pressure-head  of  200  Ibs.  per  square    inch,  the  delivery  of 
water  being  100  cu.  ft.  per  minute.      Fin.d  the  total  H.P.,  assuming  that 
the  buckets  are  so  formed  that  the  water  is  returned  parallel  to  its  origi- 
nal direction,  and  leaves  without  energy. 

If  the  actual  H.P.  is  70.3,  what  is  the  efficiency  ? 

Ans.  87.22  ;  .805. 

32.  A  vane  moves  in  the  direction  ABC  with  a  velocity  of  10  ft.  per 
second,   and  a  jet    of  water   impinges  upon  it  at  B  in  the  direction  BD 
with  a  velocity  of  20  ft.  per  second ;  the  angle  between  BC  and  BD  is 
30°.     Determine  the  direction   of  the  receiving-lip  of  the  vane,  so  that 
there  may  be  no  shock. 

Ans.  The  angle  between  lip  and  BC  —  23°  47'. 

33.  A  jet  moves  in  a  direction  ABC  with  a  velocity  v  and  impinges 
upon  a  vane  which  it  drives  in  the  direction  BD  with  a  velocity  4z/. 
The  angle  ABD  is  165°.     Determine  the  direction  of  the  lip  of  the  vane 
at  B,  so  that  there  may  be  no  shock  at  entrance. 

Ans.  The  angle  between  lip  and  direction  of  stream  =  14°  3'. 

34.  The  lip  angle  of  a  given  bucket  is  30°,  the  relative  velocity  (¥}  is 
one  half  the  velocity  (7/1)  with  which  the  water  reaches  the  lip.     If  there 
is  to  be  no  "  loss  in  shock,"  find  the  speed  (u)  of  the  bucket,  the  direc- 
tion (y]  of  the  entering  water,  and  show  that  if  the  speed  is  to  be  increased 
10  per  cent,  the  lip  angle  must  also  be  increased  by  55.6  per  cent. 

Ans.  y  —  15°  31'. 

35.  A   stream    moving  with   a   velocity  v  impinges    without    shock 
upon  a  curved  vane  and  drives  it  in  a  direction  inclined  at  an  angle  to 
the  direction  of  the  stream.     The  angle  between  the  lip  of  the  vane  and 
the  direction  of  the  stream  is  x,  and  V  is  the  relative  velocity  of  the 
water  with  respect  to  the  vane.     If  the  speed  of  the  vane  is  changed  by 
a  small  amount,  say  n  per  cent,  show  that  the  corresponding  change  in 
the  direction  of  the  lip,  in  order  that  the  water  might  still  strike  the 

n     v     . 

vane  without  shock,  is  —  -v^sm  x. 
100  V 

36.  A  jet  issues  through  a  thin-lipped  orifice  i    sq.  in.  in  sectional 
area  in  the  vertical  side  of  a  vessel  under  a  pressure  equivalent  to  a 
head  of  900  ft.  and  impinges  on  a  curved  vane,  driving  it  in  the  direc- 
tion of  the  axis  of  the  jet.     The  water  enters  without  shock  and  turns 
through  an  angle  of  60°  before  it   leaves  the  vane.     Find  (a)  the  speed 
of  the  vane  which  will  give  a  maximum  eflect ;  (b)  the  pressure  on  the 


EXAMPLES.  4*3 

vane  ;  (c)  the  work  done  ;  (d)  the  absolute  velocity  with  which  the  water 
leaves  the  vane;  (e)  the  reaction  on  the  vessel,  disregarding  contraction. 
Ans.  (a}  80  ft.  per  sec.  ;  (6)  320.9  Ibs.  ;  (c)  46.68  H.P. ;  (d)  184  ft. 
per  sec.  ;  (e)  781.25  Ibs. 

37.  A  stream  of  thickness  /  and    moving   with   the  velocity  v   im- 
pinges without  shock  upon  the  concave  surface  of  a  cylindrical  vane  of 
a  length    subtending  an  angle  2«  at  the  centre.     Determine    the  total 
pressure  upon  the  vane  (a)  if  it  is  fixed  ;  (b)  if  it  is  moving  in  the  same 
direction  as  the  stream  with  the  velocity  u.     In  case  (d)  also  find  (c)  the 
work  done  on  the  vane. 

Ans.  (a)  2--<*/va  sin  a;  (b)  2—  bt(v  —  uY  sin  a  ;  (c)  2™-blu(v  -  u?  sin3  a. 
£  &  S 

38.  A  stream  of  water,  2  sq.  ins.  in  sectional  area,  delivers  I  cu.  ft. 
per  second  against   the  concave  side   of  a  hemispherical    cup,    which 
moves  with  a  velocity  of  20  ft.  per  second  in  the  direction  of  the  jet. 

Find  the  impulse,  the  work  done,  and  the  efficiency. 

39.  A  curved  vane  subtends  an  angle  of  90°  at  the  point  of  intersec- 
tion of  the   normals  at  the   two  edges,  and   receives  without  shock  a 
stream    of  water  2  ft.  wide  and  \  in.   thick,   moving  with  a  velocity  of 
20  ft.  per  second    and  driving    the   vane    in  the    same  direction.     The 
actual  direction  of  the  water  is  turned  through  an  angle  of  45°.     Find 
(a)  the  speed  of  the  vane  ;  (b}  the  velocity  with  which   the  water  leaves 
the  vane  ;  (c)  the  total  pressure  on  the  vane;  (d)  the  efficiency. 

Ans.  (a]  10  ft.  per  sec.  ;  (b}  14.14  ft.  per  sec.;  (c)  23,017  Ibs. ;  (d)  .25. 

40.  A  vane  is  in  the  form  of  the  segment  AB  of  a  circle  subtending 
an  angle  of  120°  at  the  centre  O.     A  stream  of  water,  moving  with  a 
velocity  v\ ,  strikes  the  vane  tangentially  at  A  and  drives  it  in  the  same 
direction  with  a  velocity  n.     Find  the  velocity  (7/3)  with  which  the  water 
leaves  the  vane,  and  show  that  it  leaves  in  the  direction  OB  if  2/1  =  \\u, 
and  that  the  direction  has  turned  through  90°  if  v\  =  yi.      Find  the 
efficiency  in  the  two  cases,  and  show  that  v\  =.yt  corresponds  to  maxi- 
mum efficiency. 

Ans.  v<?  —  v?  —  $viu  +  «2;  -f  ;  |.     If  vi  =  2u,  z/a  —  u,  the  direc- 
tion turns  through  60°  and  i?  =  -f. 

41.  A  stream  of  water  of  36  sq.  ins.  section  moves  in  a  direction  ARC 
and   delivers  4  cu.  ft.  of  water   per  second   upon   a  vane    moving  in  ;i 
direction  BD  with  a  velocity  of  8  ft.  per  second,  the  angle  between  BC 
and  BD  being  30°.     Find  (a)  the  best  form  to  give  to  the  vane  ;  (b}  the 
velocity  of  the  water  as  it  leaves  the  vane;  (c)  the  mechanical  effect  of 
the  impinging  jet;  and  (d)  the  efficiency,  the  angle  turned  through  by 
the  jet  being  90°. 

Ans.  (a)  The  angle  between  lip  and  BC  —  23°  48' ;  (b)  3.088  ft. 
per  second  ;  (c)  962.8  ft.  per  second  ;  (d}  .963. 

42.  In  an  I.  F.  tangential  turbine  find  (a)  the  loss  due    to  hydraulic 
resistances,  (b)  the  useful  effect,  (c)  the  efficiency,  (d)  the  lip  angles  from. 


414  EXAMPLES. 

the  following  data  :  Q  =  i  cu.  ft.  per  second  ;  k  =  100  ft.;  /a  =  /4  =  £  ^ 
JK  —  1 5° ;  5*1  =  6ra  ;  </i  =  */a ;    and  uy  =  V*. 

Ans.  (a)  736.72  ft.-lbs.;  (b)  4866.9  ft.-lbs.;  (V)  .7787  ;  a  —  150°  33'. 
/?  -  47°  16'. 

43.  In  an  O.  F.  turbine  of  the  tangential  type  find  the  lip  angles,  the 
losses  of  head  due  to  the  velocity  (z/a)  of  the  effluent  water,  and  to  hy- 
draulic resistances,  and  also  find  the  efficiency  from  the  following  data  : 
y  —  30°;  2r,  =  r2 ;  //  =  30  ft.  ;  di  =  d* ;  ;/2  =  Fa ;  /2  =  2/4  =  .  1 25. 

yto.  a  =  123°  27';  ft  =  13°  30';   1.688  ft.;  5.243  ft.;  .769. 

44.  An  I.  F.  tangential  turbine,  with  parallel  faces  (di  =  d9)  and  an 
inlet  surface  of  6  ft.  diameter,  delivers  10.76  H.P.  under  a  head  of  150  fu 
The  direction  of  the  inflowing  stream,  which  is  5  ins.  wide,  makes  an 
angle  (y)  of  10°  with  the  turbine's  periphery,  and  the  diameters  of  the 
inlet  and  outlet  surfaces  are  in  the  ratio  of  4  to  3.     If  f.2  =/4  =  .1,  and 
if  also  it  is  assumed  that  z*a  =  K2 ,  find  (a)  the  inlet  and  outlet  lip  angles  i 
(b)  the  loss  of  H.P.  due  to  hydraulic  resistances;  (c)  the  loss  of  H.P. 
corresponding  to  the  velocity  with  which  the  water  leaves  the  turbine  ; 
(d)  the  efficiency  ;  (e)  the  quantity  of  water  passing  through  the  turbine  ; 
(f)  the  thickness  of  the  inflowing  stream  ;  (g)  the  speed  in  revolutions, 
per  minute. 

Ans.  (a)  161°  13',  40°  36' ;  (b)  1.309  ;  (c)  .708  ;  (d)  .842  ;  (e)  f  cu. 
ft.  per  second  ;  (/)  .231  in.  ;  (g)  141. 

45.  In  the  preceding  example  if,  instead  of  making  tt-t  =  F2 ,  it   is 
iiFSumed  that  the  flow  at  outlet  is  radial,  find  the  inlet  and  outlet  lip 
angles  so  that  the  efficiency  may  remain  the  same.     Also  find  the  losses 
in  H.P.  due  to  hydraulic  resistance  and  to  the  energy  carried  away  by 
the  effluent  water,  and  determine  the  speed  in  revolutions  per  minute, 
(cv*  —  {%.)  Ans.  161°  21',  33°  17' ;  1.369;  .623;  139.8. 

46.  A  stream  4  ins.  wide  and  supplying  f  cu.  ft.  of  water  per  second 
drives  an  O.  F.  turbine  of  the  tangential  type,  in  which  the  diameters 
of  the  inlet  and  outlet  surfaces  are  in  the  ratio  of  3  to  4.     The  turbine 
faces  are  parallel,  and  the  inflowing  stream  makes  an  angle  of  20°  with 
the  periphery.     The  head   is   100  ft.     First  assuming  that  it*  =  Vi ,  and 
second  that  the  outlet  flow  is  radial,  the  efficiency  being  the  same,  de-^ 
termine  (a)  the  inlet  and  outlet  lip  angles  ;  (b)  the  useful  work ;  (c)  the 
efficiency ;  (d)    the  thickness  by  the  stream  ;  (e)  the  speed   in  revolu- 
tions per  minute,  the  outer  diameter  being  5  ft.     (Disregard  hydraulic 
resistances.) 

Ans.  First,  (a)  140°,  21°  12';  (b}  4368.16  ft.-lbs.:  (c)  .932; 
(d)  .3375  in.;  (<?)  216.7.  Second,  (a)  142°  19',  21°  10';  (b)  4394.7 
ft.-lbs.;  (c)  .937;  (dO-3375*:  (*)  202.45. 

47.  Solve  the  preceding  example  when  hydraulic  resistances  are  taken 
into  account,  assuming/a  =/4  =  .1  and  c^  —  .9. 

Ans.    First,     (d)    140°,   21°    12';    (b)   3766   ft.-lbs.;    (c)   .8034; 


EXAMPLES.  415 

(<f)  .356    in.;   (e)    154.2.     Second,     (a)    141°    29',   20°    40';  (b)  3766 
ft.-lbs.;  (c)  .8034;  (</)  .356  in.;  (e)  147-79- 

48.  A  jet  turbine,  of  5^  ft.  exterior  diameter  and  with  equal  inlet  and 
outlet    depths,   passes  1890  cu.   ft.  of   waier  per  hour  under  a  head  of 
loo   ft.;   the  diameter  of  the  outlet  surface  is  twice  that  of  the  inlet, 
and  the    velocity  of  the  outlet  periphery  («a)  is  equal  to  that  of  the 
inflowing  stream  (z/i),  which  is  radial  in  direction.      Find  (a)  the  useful 
effect  in  horse-power  ;  (b)  the  efficiency  ;  (c)  the  speed  in  revolutions  per 
minute,  first  disregarding  hydraulic  resistances  and  second  taking  these 
resistances  into  account.     (_/»  =  2/a  =  .2  and  cj  =  yf.) 

Ans.  'First,     (a)  3.85  H.P.;  (b)  .645;  (c)  300.    Second,     (a)  2.063 
H.P.;  (b)  .3458;  (c)  286.04. 

49.  In  the  preceding  example  how  much  water  must  the  turbine  pass, 
when  hydraulic  resistances  are  taken  into  account,  to  give  the  delivery 
of  4  H.P.?  Ans.  1.017  cu-  ft.  per  second. 

50.  A  centrifugal  outward-flow  turbine  with  equal  inlet   and  outlet 
depths  and  working  under  the  head  of  200  ft.  over  the  inlet  passes  i  cu. 
ft.  of  water  per  second.    The  angle  y  —  \  5°;  5n  =  4^2  ;  and  the  velocity  at 
outlet  is  radial,  i.e.,  d  =  90°.   Find  (a)  the  peripheral  speed  ;  (b)  the  lip  angle 
at  inlet;  (c)  the  lip  angle  at  outlet;  (d)  the  areas  at  inlet  and  outlet; 
(e)  the  efficiency,  taking^a  =f*  =  .125. 

Ans.  (a)  55.215  ft.  per  second;  (£)  157°  18';  (c)  18°  40';  (//)  .3623 
sq.  ft.,  .0428  sq.  ft.;  (e)  .778. 

51.  A  centrifugal   inward-flow  turbine  with  an  efficiency  of  80  per 
cent  and  working  under  the  head  of  200  ft.  passes  i  cu.  ft.  of  water  per 
second.      The   angle  y  —  15°;  4^1  =  5r2 ;  and  ^2  =   Ka.       Find  (a)  the 
peripheral  speed  ;  (b)  the  lip  angles  at  inlet  and  outlet ;  (c)  the  inlet  and 
outlet  areas;    (d)  the  useful  work,  taking  /a  =/4  =  .125. 

Ans.  (a)  55.215  ft.  per  second;  (b)  157°  18',  32°  28';  (c)  .03623 
sq.  ft.,  .0369  sq.  ft.;  (d)  8888f  ft.-lbs. 


CHAPTER    VI. 
'      VERTICAL   WATER-WHEELS. 

1.  Classification    of   Water- wheels. — Water-wheels    are 
large  vertical  wheels  which  are  made  to  turn  on  a  horizontal 
axis  by  water  falling  from  a  higher  to  a  lower  level.      These 
wheels  may  be  divided  into  three  classes : 

(a)  Undershot  Wheels,  in  which  the  water  is  received  near 
the  bottom  and  acts  by  impulse. 

(7>)  Breast  Wheels,  in  which  the  water  is  received  a  little 
below  the  axis  of  rotation  and  acts  partly  by  impulse  and  partly 
by  its  weight. 

(e]  Overshot  Wheels,  in  which  the  water  is  delivered  nearly 
at  the  top  and  acts  chiefly  by  its  weight. 

2.  Undershot  Wheels. — Wheels  of  this  class,  with  plane 
floats  or  buckets,  are  simple  in  construction,  are  easily  kept  in 
repair,  and  were  in   much  greater  use  formerly  than  they  are 
now.      They  are  still  found  in  remote  districts  where  there  is 
an  abundance  of  water-power,  and  are  also  employed  to  work 
floating  mills,  for  which  purpose  they  are  suspended  in  an  open 
current  by  means  of  piles  or  suitably  moored  barges.      They 
are  made  from    10  to  25  ft.  in   diameter,  and  the  floats,  which 
are  from  24  to   28  ins.  deep,  are  fixed  either  normally  to  the 
periphery  of  the  wheel,    or  with    a   slight  slope   towards   the 
supply-sluice,   the  angle  between  the  float  and    radius  being 
from  15°  to  30°.      The  depth  of  a  float  is  from  one  fourth  to 
one  fifth  of  the  radius  and  should  not  be  less  than  from    12   to 
14  ins.    They  are  from  14  to  16  ins.  apart,  and  generally  from 

416 


UNDERSHOT  WHEELS.  4*7 

•one  half  to  one  third  of  the  total  depth  of  float  is  acted  upon 
by  the  water. 

Let  Fig.  236  represent  a  wheel  with  plane  floats  working 
in  an  open  current. 


FIG.  236. 

Let  i\  be  the  velocity  of  the  current. 
Let  u  be  the  velocity  of  the  wheel's  periphery. 
Let  Q  be  the  delivery  of  water  in  cubic  feet  per  second. 
The  water  impinges  upon  a  float,  is  reduced  to  relative  rest, 
and  is  carried  along  with  the  velocity  u.      Thus 


and 


wQ 

the  impulse  —  — (z/t  —  u)t 
<s 


the  useful  work  per  second  =  —  u(vv  —  u). 

<5 


Hence 


the  efficiency  = 


which  is  a  maximum  and  equal  to  -  when  u  =  -7/r 


41 8  UNDERSHOT   WHEELS  IN  A  STRAIGHT  RACE. 

Theoretically,  therefore,  the  wheel  works  to  the  best 
advantage  when  the  velocity  of  its  periphery  is  one  half  of  the 
current  velocity.  Even  then  its  maximum  theoretic  effect  is 
only  50  per  cent,  and  in  practice  this  is  greatly  reduced  by 
frictional  and  other  losses,  so  that  the  useful  effect  rarely 
exceeds  30  per  cent.  Undershot  wheels  with  plane  floats  are 
cumbrous,  have  little  efficiency,  and  should  not  be  used  for 
falls  of  more  than  5  ft. 

Again,  let  A  be  the  water-area  of  a  float,  and  w  be  the 
specific  weight  of  the  water. 

wQ  is  somewhat  less  than  wAi\  ,  as  there  will  be  an  escape 
of  water  on  both  sides  of  the  float. 

Let  wQ  —  kwAv^,  k  being  some  coefficient  (<  i)  to  be 
determined  by  experiment.  Then 

the  useful  work  per  second  —  kAw—(i\  —  //), 

cb 

kA 

and  its  maximum  value  =  — — v*w. 

4g 

According  to  Bossut's  and  Poncelet's  experiments  a  mean 

4  2 

value  of  k  is  -,  and  the  best  effect  is  obtained  when  //  =  ~vl  > 

^  A     f~f>/4'~i  ^ 

the  corresponding  useful  work v  being-  ^   and   the  effi- 

D  <S 

48 

ciency . 

125 

3.  Wheels  in  Straight  Race. — Generally  the  water  is  let 
on  to  the  wheel  through  a  channel  made  for  the  purpose,  and 
closely  fitting  the  wheel,  so  as  to  prevent  the  water  escaping 
without  doing  work.  For  this  reason,  also,  the  space  between 
the  ends  of  the  floats  in  their  lowest  positions  and  the  channel 
is  made  as  small  as  is  practicable  and  should  not  exceed  2  ins^ 
Hence  k,  and  therefore  also  the  efficiency,  will  be  increased.. 


UNDERSHOT   H/HEELS  IN  A  STRAIGHT  RACE. 


4*9 


Assume  the  channel  to  be  of  a  uniform  rectangular  section  and 
to  have  a  bed  of  so  slight  a  slope  that  it  may  be  regarded  as 
horizontal  without  sensible  error. 

The  wheel  is  usually  from  24  to  48  ft.  in  diameter,  with  24 
to  48  floats,  either  radial  or  inclined.  The  floats  are  12  to 
2O  ins.  deep,  or  about  2^  to  3  times  the  depth  of  the  approach- 
ing stream.  The  fall  should  not  exceed  4  ft.  Let  the  floats 
be  radial,  Fig.  237. 


FIG.  237. 

Let  kl  be  the  depth  of  the  water  on  the  up-stream  side  of 

the  wheel. 
Let  h2  be  the  depth  of  the  water  on  the  doivn-stream  side 

of  the  wheel. 
Let  b  be  the  width  of  the  race.      Then 

bh^vl  —  Q  =  bhji. 

The  impulse  =^  impulse  due  to  change  of  velocity 

+  impulse  due  to  change  of  pressure 

WQ,  x  W^(l   2  2 

—   g  (V\      u)-r  2  v  i  •-   2) 


420  EXAMPLE. 

and  the  useful  work  per  second 


The  total  available  work  =  —  -—  -*-. 

S      2 

Hence  the  efficiency          -  ^  -  u)  +  f^  -  - 

^  I  11  it 

The  second  term  is  negative,  since  //2  >  /^  ,  and  the  maxi- 
mum theoretic  efficiency  may  be  easily  shown  to  be  <  .5. 

Ex.  An  undershot  wheel  with  straight  floats  and  weighing  15,000 
Ibs.  works  in  a  rectangular  channel  with  horizontal  bed  and  of  the  same 
-width  as  the  wheel,  viz.,  4  ft.;  the  stream  delivers  28  cu.  ft.  of  water  per 
second,  and  the  efficiency  of  the  wheel  is  £.  Find  the  relation  between 
the  up-stream  (v\)  and  down-stream  (»)  velocities. 

If  the  up-stream  velocity  is  20  ft.  per  second,  find  the  down-stream 
velocity.  If  the  diameters  of  the  wheel  and  bearings  are  20  ft.  and  4 
ins.,  respectively,  and  if  the  coefficient  of  friction  is  .008,  determine  the 
mechanical  effect. 

28  =  4-#i?/i 


or  h\  —  —      and     //,  =  -. 

z/i  u 

Therefore  the  efficiency  =  —  (vi  —  «)  +  ^(-^  —  —.  }  =  - 
*       s>i*v  7'i2\?V      «V       3 

2t(  f  2/i    +  U\  I 

—  i(Vi  —  »)[  I   —    112  -  r-r]  =  -. 

Vi          ,       \  vSu*  I       3 

If  vi  —  20  ft.  per  second,  then 

;/  /  7    20  +  «\        i 

—  (20  —  «)(  I   --  —       ——)=-. 

200  \         25       H      /        3 

It  is  found  by  trial  that  u  lies  between  5.9  and  6  and  is  very  approxi- 
mately 5.97  ft.  per  second. 

The  total  available  power  =    -    .  28  .  —  =  10937.5  ft.  -Ibs.  per.  sec. 

Therefore  the  actual  mechanical  effect  =  -(10937.5) 

=  3645.83  ft.-lbs.  per  sec. 
The  work  absorbed  by  bearing  friction  =  .008  x  15000  x  5.97  x  — 

=  11.94  ft.-lbs.  per  sec. 
The  net  delivery  in  ft.-lbs.  =  3645.83  -  11.94  =  3633.89- 


LOSSES. 


421 


Losses.—  Four  principal  losses  may  be  considered,  viz.  : 
(i)   The  loss  of  Ql  cubic  feet  of  the  deeper  fluid  elements 
which  do  not  impinge  upon  some  of  the  foremost  floats. 
According  to  Gerstner, 


»t  being  the  number  of  the  floats  immersed,  and  c  being  -J  or 
$  according  as  the  bottom  of  the  race  is  straight  or  falls 
abruptly  at  the  lowest  point  of  the  wheel. 

(2)  The  loss  of  Q2  cubic  feet  of  water  which  escapes  between 
the  wheel  and  the  race-bottom. 

Approximately,  the  play  at  the  bottom  may  be  said  to  vary 
from  a  minimum,  sl  —  BC,  when  a  float  AB  is  in  its  lowest 
position,  Fig.  238,  to  a  maximum,  BlCl  —  CD  —  £>2C2,  when 


c 
FIG.  238. 


tv 


two  floats  A^B^  ,  AZBZ  are  equidistant  from  the  lowest  position,, 
Fig.  238.      Thus  the  mean  clearance 


+  BD)  = 


,  nearly, 


being  the  wheel's  radius. 


422  LOSSES. 

2  Tfy 

But  -  !  =  distance  between  two  consecutive  floats 
n 

=  2  .  BVD,  very  nearly, 
n  being  the  total  number  of  floats.      Hence 


and  therefore  the  mean  clearance  =  s.  -I  —   —~. 

4    ;/3 

Again,  the  difference  of  head  on  the  up-stream  and  down 
stream  sides 


and  the  velocity  of  discharge,   frf,   through  the    clearance  is 
given  by  the  equation 


Hence 

Introducing  .7  as  a  coefficient  of  hydraulic  resistance, 

I    7T2 

'    4    n 

If  the  depth  of  the  stream  is  the  same  on  both  sides  of  the 
wheel,  i.e.,  if  hv  =  7/2,  then 

I'd  =  i'r 

(3)  The  loss  of  <23  cubic  feet  of  water  which  escapes  between 
the  wheel  and  the  race-sides. 

Let  s2  be  the  clearance  on  each  side.      Then 

<2H  =  .7  X  2//tJ2?v  =  i-4^iVV» 
,7  being  a  coefficient  of  hydraulic  resistance. 


MECHANICAL   EFFECT.  42$ 

(4)   Finally,  if  W\bs.  is  the  weight  on  the  wheel-journals, 
the  loss  due  to  journal  friction 


yu  being-  the  journal  coefficient  of  friction,  and  p  the  journal 
radius. 

Actual  Delivery.  —  Thus  the  actual  delivery  of  the  wheel  in 
foot-pounds 


Remarks.  —  These  wheels  are  most  defective  in  principle, 
as  they  utilize  only  about  one  third  of  the  total  available 
energy.  They  may  be  made  to  work  to  somewhat  better 
advantage  by  introducing  the  following  modifications: 

(a]  The  supply  may  be  so  regulated  by  means  of  a  sluice- 
board  that  the  mean  thickness  of  the  impinging  stream  is 
about  6  or  8  ins.  If  the  thickness  is  too  small,  the  relative 
loss  of  water  along  the  channel  will  be  very  great.  If  the 
thickness  is  too  great,  the  floats,  as  they  emerge,  will  have  to 
raise  a  heavy  weight  of  water.  The  sluice-board  is  inclined 
at  an  angle  of  30°  to  40°  to  the  vertical,  so  that  the  sluice- 
opening  may  be  as  near  the  wheel  as  possible,  thus  diminishing 
the  loss  of  head  due  to  channel  friction,  and  is  rounded  at  the 
bottom  to  prevent  a  contraction  of  the  issuing  fluid.  Neglect- 
ing frictional  losses,  etc., 

r  i    cc  ^(  TT  ,    v\         *2\        (  Ioss    °f    energy 

the  useful  effect  =  wQ\H  -4-  —  --  1  —  1  , 

ig       tgi        \  due  to  shock 

l._^}_^Q^^f. 

2g         2gl 


H  being  the  difference  of  level  between  the  point  at  which  the 
water  enters  the  wheel  and  the  surface  of  the  water  in  the  tail- 


424  MECHANICAL   EFFECT. 

race,  i.e.,  the  fall.  H  is  usually  very  small  and  may  be  nega- 
tive. 

If  the  vanes  are  inclined,  the  resistance  to  emergence  is 
not  so  great,  and  the  frictional  bed  resistance  between  the 
sluice  and  float  is  practically  reduced  to  nil.  With  a  straight 
bed  and  small  slope  (i  in  10)  the  minimum  convenient  diameter 
of  wheel  is  about  14  ft. 

(b)  The  bed  of  the  channel  for  a  distance  at  least  equal  to 
the  interval  between  two  consecutive  vanes  may  be  curved  to 
the  form  of  a  circular  arc  concentric  with  the  wheel,  with  the 
view  of  preventing  the  escape  of  the  water  until  it  has  exerted 
its  full  effect  upon  the  wheel.  When  the  bed  is  curved,  the 
minimum  convenient  diameter  of  wheel  is  about  10  ft. 

An  undershot  wheel  with  a  curb  is  in  reality  a  lo\v  breast 
wheel,  and  its  theory  is  the  same. 

(<:)  The  down-stream  channel  may  be  deepened  so  that  the 
velocity  of  the  water  as  it  flows  away  becomes  >  vr  The 
impulse  due  to  pressure  is  then  positive,  which  increases  the 
useful  work  and  therefore  also  the  efficiency. 

(d*)  The  down-stream  channel  may  be  widened  and  a  slight 
counter-inclination  given  to  the  bed.  What  is  known  as  a 
standing-wave  is  then  produced,  in  virtue  of  which  there  is  a 
sudden  rise  of  surface-level  on  the  down-stream  side  above 
that  on  the  up-stream  side.  This  allows  of  the  wheel  being 
lowered  by  an  amount  equal  to  the  difference  of  level  between 
the  surfaces  of  the  standing-wave  and  of  the  water-layer  as  it 
leaves  the  wheel,  thus  giving  a  corresponding  gain  of  head. 

(e)  The  introduction  of  a  sudden  fall  has  been  advocated 
in  order  to  free  the  wheel  from  back-water,  but  it  must  be 
borne  in  mind  that  all  such  falls  diminish  the  available  head. 

4.  Poncelet  Wheel  (Figs.  239,  240). —  Thus  undershot 
wheels  with  flat  buckets  have  a  small  efficiency  because  of 
the  loss  of  energy  in  shock  at  entrance  and  because  of  the 
loss  of  energy  carried  away  'by  the  water  oh  leaving  the: 
wheel.  These  losses  have  been  considerably  modified  in 


PONCE  LET  WHEEL. 


425 


Poncelet's  wheel,  which  is  often  the  best  motor  to  adopt  when 
the  fall  does  not  exceed  6J  ft.,  and  which,   in  its  design,  is 


FIG.  239. 


m// 


FIG.  240. 

governed  by  two  principles    that    should  govern  every  perfect 
water-motor,  viz.  : 

(i)    That  the  loss  of  energy  in  shock  at  entrance  shotild  be- 
a  minimum. 


426  PONCE  LET  WHEEL. 

(2)  That  the  velocity  of  the  water  as  it  leaves  the  wheel 
should  be  a  minimum. 

The  vanes  are  curved  and  are  comprised  between  two 
crowns,  at  a  slightly  greater  distance  apart  than  the  vane- 
width  ;  the  inner  ends  of  the  vanes  are  radial,  and  the  water 
acts  in  nearly  the  same  manner  as  in  an  impulse  turbine. 

A  Poncelet  wheel  of  from  10  to  13  ft.  in  diameter  has  36 
floats,  while  for  wheels  of  from  20  to  23  ft.  in  diameter  the 
number  of  floats  is  about  48.  The  wheels  are  usually  from  10 
to  20  ft.  in  diameter  and  have  from  32  to  48  floats  which  may 
be  of  plate-iron  or  wood. 

First.  Assume  that  the  outer  end  of  a  vane  is  tangential 
to  the  wheel's  periphery,  that  the  impinging  layer  is  infinitely 
thin,  and  that  it  strikes  a  float  tangentially. 

Let  of  (Fig.  241)  be  a  float,  and  aq  the  tangent  at  a. 

The  velocity  of  the  water  relatively 
to  the  float  =  ?',  —  u. 

The  water,  in  virtue  of  this  velocity, 
ascends     on    the    bucket    to    a    height 

/  \2 

pq  =  — — ,  then  falls  back  and  leaves 

the  float  with  the  relative  velocity  v^  —  u  and  with  an  absolute 
velocity  v^  —  2u.  This  absolute  velocity  is  nil  when  the  speed 
of  the  wheel  is  such  that  u  =  -J-z'j ,  and  the  theoretical  height 


2 


of   a   float   is   pq  =  —  -1-.      The   total    available   head  is   thus 
42£- 

changed  into  useful  work,  and  the  efficiency  is  unity,  or  perfect. 

Taking  R  as  the  mean  radius  of  the  crown  and  um  as  the 

corresponding  linear  velocity,   the  mean  centrifugal  force  on 

u  * 

each  unit  of  fluid  mass  is  -^-  and  acts  very  nearly  in  the  direc- 

K 

tion  of  gravity,  so  that  the  height  pq  of  a  float  may  be  approxi- 
mately expressed  in  the  form 

I™ 

pq~- 


PONCE  LET   WHEEL. 


427 


V  being  the  velocity  with  which  the  water  commences  to  rise 
on  the  float. 

Practically,  however,  the  float  is  not  tangential  to  the 
periphery  at  a,  as  the  water  could  not  then  enter  the  wheel. 
Also,  the  impinging  water  is  of  sensible  thickness,  strikes  the 
periphery  at  some  appreciable  angle,  and  in  rising  and  falling 
on  the  floats  loses  energy  in  shocks,  eddies,  etc. 

Let  the  water  impinge  in 
the  direction  ac,  Fig.  242, 
and  take  ac  =  vr 

Take  ad  in  the  direction 
of  and  equal  to  ?/,  the 
.velocity  of  the  wheel's  pe- 
riphery. 

Complete  the  parallelo- 
gram bd. 

Then  cd  =  ab  =  V is  the 
velocity  of  the  water  rela- 
tively  to  the  float. 


C| 

FIG.  242.  FIG.  243. 

That  there  may  be  no  shock  at  entrance,  ab  must  be  a 
tangent  to  the  vane  at  a. 

Again,  the  water  leaves  the  vane  in  the  direction  of  ba 
produced,  and  with  a  relative  velocity  ac  =  ab  =  V. 

Complete  the  parallelogram  de.  Then  ag  (=  v2)  is  the 
absolute  velocity  of  the  water  leaving  the  wheel. 

Evidently  cdg  is  a  straight  line. 

Let  the  angle  cad  =  y,  and  the  angle  bad  =  n  —  a. 

From  the  triangle  adc, 


V'2  = 


—  2v4  cos 


(0 


428  PONCE  LET  WHEEL 


v*  =   F2  +  u*  —  2  Vu  cos  a  ;   .      .      .      ,      (2) 
Sm 


vl        sn  a 


From  the  triangle 

v*  =  V*  +  u*  +  2  Vu  cos  a  .....     (4) 
By  equations  (i),  (2),  and  (4), 


COS         — 


Therefore  the  useful  work  per  second 

iv  Q 

= 2u(v.  cos  y  —  u} (5) 

g 

ivQ  v  2  cos2  Y 
This    is    a    maximum   and    equal    to  -   when 

v   cos  Y 
u  =  -  - — — ,  and  the   maximum  efficiency  is  cos2  y.      Hencefc 

too,  the  angle  adb  —  90°,  and,  by  Fig.  243, 

bd       2pd 

tan  (TT  —  a)  =  — -  —  — T  =  2  tan  y.        .      .      (6) 
ad        ad 

Also, 

V       ab 


The  efficiency  is  perfect  if  y  is  nil,  and  therefore  a  =  180°. 
Practically  this  is  an  impossible-  value,  but  the  preceding  cal- 
culations indicate  that  y  should  not  be  too  large  (usually 
<  30°),  and  that  the  speed  of  the  wheel  should  be  a  little  less 
than  one  half  of  the  velocity  of  the  inflowing  stream. 

Take  y  =  15°  as  a  mean  value.      Then 

u  —  fj  X  -484,  and  the  efficiency  =  .993. 


PONCE  LET   WHEEL. 


429 


The  best  practice  indicates  the  relation  112^=  2Ou.  It 
must  be  borne  in  mind  that  the  theory  applies  to  one  elemen- 
tary layer  only,  say  the  mean  layer,  and  that  all  the  other 
layers  enter  the  wheel  at  angles  differing  from  15°,  thus  giving 
rise  to  "  losses  of  energy  in  shock."  The  losses  of  energy  in 
frictional  resistance,  eddy  motion,  etc.,  in  the  vane-passages 
have  also  been  disregarded.  Tangential  entrance  is  not  possi- 
ble in  practice  and  the  efficiency  does  not  exceed  .65  for  falls 
up  to  4  ft.,  is  .60  for  falls  of  from  4  to  5.5  ft.,  and  is  from  .55 
to  .50  for  falls  of  from  5.5  to  6.5  ft.  The  greater  efficiency  of 
the  Poncelet  wheel,  as  compared  with  wheels  having  flat 
buckets,  very  clearly  shows  the  importance  of  bringing  the 
water  on  to  the  wheel  in  such  a  manner  as  to  avoid  loss  of 
energy  in  shock  and  in  the  production  of  eddies.  The  layers 
of  water,  flowing  to  the  wheel  under  an  adjustable  sluice  and 
with  a  velocity  very  nearly  equal  to  that  due  to  the  total  head, 
may  be  all  made  to  enter  at  angles  approximately  equal  to 
15°,  and  the  corresponding  losses  in  shock  reduced  to  a  mini- 
mum by  forming  the  course  as  follows: 

The  first  part  of  the  course  FG,  Fig.  244,  is  curved  in  such 


FIG.  244. 

a  manner  that  the  normal  pqr  at  any  point  /  makes  an  angle 
of  15°  with  the  radius  oq.  The  water  moves  sensibly  parallel 
to  the  bottom  FG,  and  therefore  in  a  direction  at  right  angles 


43°  PONCE  LET   WHEEL. 

to  pr.  Hence  at  q  the  direction  of  motion  makes  an  angle  of 
15°  with  the  tangent  to  the  wheel's  periphery.  If  or  is  drawn 
perpendicular  to  pr,  then  or  —  oq  sin  15°  =  a  constant. 

Thus  the  normal  pqr  touches  at  r  a  circle  concentric  with 
the  wheel  and  of  a  certain  constant  diameter. 

The  initial  point  F  of  the  profile  FG  is  the  point  in  which 
the  tangent  to  this  circle,  passing  through  the  upper  edge  of 
the  sluice-opening,  cuts  the  bed  of  the  supply-channel. 

Let  d  be  the  depth  of  the  crown  or  shrouding,   i.e.,   the 
normal    distance   between   the  outer   and   inner 
peripheries  of  the  wheel. 
Let  b  be  the  width   and  /  the  thickness  of  the    sheet  of 

water  entering  the  wheel. 

Then,  disregarding  the  thickness  of  the  floats,  the  capacity 
of  the  portion  of  the  wheel  passing  in  front  of  the  entering 
stream  per  second  is  approximately  bdum.  Practically,  the 
whole  of  this  space  cannot  be  occupied  by  the  water  and 

mbdum  =  Q  =  btv^  , 
m  being  a  coefficient  varying  from  \  to  |-. 

Thus  /,  the  thickness  of  the  stream,  =  md— 

vi 

,R  u 

=  ma  -- 

ri  vi 
If  the  efficiency  is  a  maximum,  vl  cos  y  =  2u,  and  then 

m   R 

t  =  —d  —  cos  y. 
2     rl 

The  head  over  the  mean  water  layer  at  the  point  of 
entrance 


H  being  the  available  fall.      Hence 


PONCELET   WHEEL. 


43  r 


an    average    value    of  cv    being    .9,    and   if,    as   according   to 
Grashof,  H=  i6t, 


32 


Morin  makes  the  radius  (r^)  of  the  wheel  from  two  to  three 
times  the  depth  (d)  of  the  crown,  and  Poncelet  considers  that 


IT  TT 

this  depth  should  be  about  —  and  not  less  than  —  . 

3  4 


In  order, 


indeed,  to  prevent  the  water  from  rising  over  the  top  of  the 
floats,  d  should  be  from  —  to  -//,  and  therefore  rl  from  //to 

2//,  the  latter  being  often  adopted  in  practice. 

The  area  of  the  sluice-opening  usually  varies  from  1.25^ 
to  \.^bt. 

The  inside  width  of  the  wheel  is  about  (b  -f-  -J)  ft. 

If  \  is  the  angle  subtended  at  the  centre  O  of  the  wheel 
by  the  water-arc  between  the 
point  of  entrance  a  and  the 
lowest  point  C,  Fig.  245,  of  the 
wheel,  and  if  aq'  is  drawn  hori- 
zontally, then  Aq'  is  approxi- 
mately the  height  of  the  float, 
and  the  theoretic  depth  ^  of  the 
crown  is  given  by 

d  =  AC  —  Aq'  +  Cq' 
=  Ag'  +  OC-  Oq' 

V2 
=  *  4-  r,(i  —  cos 


In  practice  it  is  usual  to  increase  this  depth  by  ^,  the  thick- 
ness of  the  impinging  water-layer,  and  therefore 


432 


EFFICIENCY  OF  PONCE  LET   WHEEL. 


-cos 


The  buckets  are  usually  placed  about  I  ft.  apart,  measured 
along  the  circumference,  but  the  number  of  the  buckets  is  not 
a  matter  of  great  importance.  There  are  generally  36  buckets 
in  wheels  of  10  to  14  ft.  diameter,  and  48  buckets  in  wheels 
of  20  to  23  ft.  diameter. 

It  may  be  assumed  that  the  water-arc  is  equally  divided  by 
the  lowest  point  C  of  the  wheel,  so  that 

the  length  of  the  water-arc  =  2\rl  =  2uT, 

T  being  the  time  of  the  ascent  or  descent  of  the  water  in  the 

bucket. 

In  the  middle  position,  the  upper  end  of  the  bucket  should 

be  vertical,  and  if  the  float  is  in 
the  form  of  a  circular  arc,  its  radius 
;•'  =  d  sec  (n  —  a),  a  being  the 
angle  between  the  bucket's  lip  and 
the  wheel's  periphery. 

The  time  of  ascent  or  descent  is 
also  given  by 


_  9^  +_smjf>     I 

i6    V 


where  sin  —  =  -/cos  (n  —  a]. 

2  x  ' 

5.  Efficiency  corresponding  to   a  Minimum  Velocity  of 
Discharge  (#2). — From  Fig.  242, 

ao  (=  \ag)       £(V2) 


sin  Y 
sin  aW 


ad 


Hence  for  any  given  values  of  u  and  y,  v^  is  a  minimum 
when  sin  aod  is  greatest,  that  is,  when  aod  =  90°,  or  ag  is  at 


EXAMPLE.  433 

right  angles  to  de.      Then  also  ad  —  ae  =  ab,  or  u  =  F,  and 
ac  bisects  the  angle  bad.      Thus 

z\  =  2u  cos  y     and     v^  =  2u  sin  ^. 
The  useful  work 
W     v 


=    —  2Z/2  COS  2V  =   - 
"  " 


<£• 

The  total  available  work 


Therefore  the  efficiency,  77,  ==  -  x—  -  , 

cos3 

and  the  H.P.  of  the  wheel         = 


550 

Experience  indicates  that  the  most  favorable  value  for  u 
lies  between  .57^  and  .6?^,  and  that  the  average  value  of  the 
•efficiency  is  about  60  per  cent. 

Although,  under  normal  conditions  of  working,  the  effi- 
ciency of  a  Poncelet  wheel  is  a  little  less  than  that  of  the  best 
turbines,  the  advantage  is  with  the  former  when  working  with 
a  reduced  supply. 

Ex.  To  design  a  Poncelet  wheel  for  a  fall  of  4*  ft.  and  a  water-supply 
of  24  cu.  ft.  per  second,  taking,  as  a  first  approximation,  y°  =  A°  =  20°. 
Mean  velocity  (v\)  at  point  of  admission  : 


vi  =  -9y  ig  -  4i  •       =  I5- 


°329  ft.  per  sec. 


speed  of  periphery  : 
Lip  angle  on : 


u  =  -i/i  cos  20°  =  7.06318  ft.  per  sec. 


tan  (it  —  a)  =  2  tan  20°  =  .728, 

and  it  —  a.  =  36°  3',     or     a  =  143°  57'. 

Value  of  iff : 

-L  

sin  -  =  Vcos  36°  3'  =  .89917, 
and  V>°  =  i28".6'  =  128°. i. 


434  EXAMPLE. 

Relative  velocity  (  V}  at  admission: 

V  =  u  sec  36°  3'  =  8.7361  ft.  per  sec. 
Value  of  r\.     Taking,  as  a  first  approximation, 

R  =  r\  =  $dt  Ui  =  u,  and  A°  =  20°,  then 
r'  =  d  sec  36°  3'  =  r\  x  .4123,  and 
128.1 


\.  sin  128°  6 


20  y"    ,80  /     n  x  .4123 

* 1 8or '  ~7b~~       ~ y  ~ (TT^Tsp ' 

which  gives  n  —  7.445  ft-»  or»  saY»  7^  ft- 

Depth  (d)  of  crown.-    Taking,  as  an  approximation,  u\=u  and  R=r\ 


=  T-7755  +  /  =  1.8  ft.,  suppose. 
More  correct  radius  of  float : 

r'  =  1.8  sec  36°  3'  =  2.226  ft. 
Values  of  R  and  u\  : 

/e  =  7. 5 --(i. 8)  =  6.6  ft. 

«,  =  — u  =  —7. 06318  =  6,2156  ft.  per  sec. 


More  correct  value  of  A  ; 

^128.1 

1 80 
A  .  7$  =  7.06318 — 


sin  1 28°  6'         — 


/— 75 

V  "     <6-; 

V   32  +  - 


'6  A/  32    ,    <6-2I56)a 

or  A  =  .298479, 

and  A°  =  17°.!. 

Thickness  (/)  ^?/"  stream  : 

1  1.8  6.6 

/  =   -      —  COS    20°  .   =    .372  ft. 

2  2  7.5 

Width  (b)  of  wheel: 

b  = =  4.29  ft. 

.372  x    15.0329 

Time  ( T]  of  ascent  or  descent  of  water  on  float : 

T=^  =  .298479^^8  =  -317  sees. 

Number  of  floats  (N\     If  spaced  i  ft.  apart, 

N  =  2Tt  .  7!  =  471,  or,  say,  48. 


FORM   OF  BUCKET. 


435 


Theoretical  maximum  power  of  wheel 

—  -^24(7.063  1  8)  2  =  4679.6  ft.-lbs.  per  sec. 

Total  available  power 

=  62^  .  24  .  4^        =  6750  ft.-lbs.  per  sec. 


Efficiency     = 


=  .693. 


6.  Form  of  Bucket.  —  The  form  of  the  bucket  is  arbitrary, 
and  may  be  assumed  to  be  a  circular  arc.  In  practice  there 
are  various  methods  of  tracing  its  form. 

METHOD  I  (Fig.  247).  The  tangent  am  to  the  bucket  at 
a  makes  a  given  angle  a  with  the  tangent 
at  a  to  the  wheel's  outer  periphery.  The 
radius  of  is  also  a  tangent  to  the  bucket 
at/.  If  the  angle  aofis  known,  the  posi- 
tion of  /on  the  inner  periphery  is  at  once 
fixed,  and  the  form  of  the  bucket  can  be 
easily  traced. 

Let  the  angle  aof  =  ,r.  Join  of  and 
let  the  tangents  to  the  bucket  at  a  and 
f  meet  in  m.  Then 

the  angle  oam  =  a  —  90°, 

"      onia  —  1  80°  •-  oam  —  aom  —  270°  —  a  —  x, 
"      mfa  —  the  angle  maf  =  J(i8o°  —  fmd) 


FIG.  247. 


a 


X 


-45°. 


Let  rx ,  r2  be  the  radii  of  the  outer  and  inner  peripheries  of 
the  wheel.      Then 


rl       oa       sin  ofa       sin  mfa 
r2~  of~~  sin  oaf  ~~   sin  oaf 


sin 


a  —  x  \ 

m(— -45J 


since  the  angle  oaf '•==  oam  —  maf  '=  -        45 


FORM   OF  BUCKET. 


Hence 


(a  x\  la  x\ 

--  45    -f-|  —  sin    --  45°  —  -] 
_,  _  f  _  2  ~^z)  \2       ^         2) 


r'  r*  sin£- 


45    - 

2  2 


X 

tan  - 


(a  x\ 

in(-  —  45°  -- 

\2  / 


tan    -  — 


ah  equation  giving  x. 

The  point  o  in  which  the  perpendicular  o'f  to  of  meets  the 
perpendicular  o' a  to  am  is  the  centre  of  the  circular  arc  required, 
and  o'f  (—  <?'#)  is  the  radius. 

METHOD  II  (Fig.  248).  Take  mad—  150°,  and  in  ma 
produced  take  ak  =  of.  With  k  as  centre  and  a  radius  equal 
to  ao  describe  the  arc  of  a  circle  intersecting  the  inner  periphery 
in  the  point  f.  Join  kf,  of,  and  af.  The  two  triangles  aof 
and  akf  are  evidently  equal  in  every  respect,  and  therefore  the 


FIG.  249. 

angle  kaf  is  equal  to  the  angle  of  a.  Drawing  ao'  at  right 
angles  to  ak  and/t/  tangential  to  the  periphery  at/,  the  angle 
0'af(=  kaf  —  90°)  is  equal  to  the  angle  o'f  a  (•=  of  a  —  90°), 
and  therefore  o'a  =  <?'/.  Thus  0'  is  the  centre  of  the  circular 
arc  required,  and  'o'a  (=  o'f)  is  the  radius. 


SLUICES. 


METHOD  III  (Fig.  249).  Let  the  bed  with  a  slope  of,  say, 
I  in  10  extend  to  the  point  c,  and  then  be  made  concentric 
with  the  wheel  for  a  distance  cc  subtending  an  angle  of  30° 
at  the  centre  of  the  wheel.  Let  the  mean  layer,  half  way 
between  the  sloping  bed  and  the  surface  of  the  advancing 
water,  strike  the  outer  periphery  at  the  point  f.  Draw  fk 
making  an  angle  of  23°  with  of,  and  take/£  equal  to  one  half 
or  seven  tenths  of  the  available  fall,  k  is  the  centre  of  the 
circular  arc  required,  and  kf\s  its  radius. 

7.  Sluices. — The  water  is  rarely  admitted  to  the  wheel 
without  some  sluice  arrangement,  which  may  take  the  form  of 
an  overfall  sluice  (Fig.  250), 
an  underflow  sluice  (Fig.  251), 
or  a  bucket  or  pipe  sluice 
(Fig.  252). 

The  pipe  sluice  is  especially 
adapted  for  a  varying  supply, 
being  provided,  for  a  certain 
vertical  distance,  with  a  series 
of  short  tubes,  so  inclined  as 
to  insure  that  the  water  enters 
the  wheel  in  the  right  direc- 
tion. Taking  .85  as  the  mean 
coefficient  of  hydraulic  resist- 
ance for  these  tubes,  the  head 
hl  required  to  produce  the 
velocity  of  entrance  vl  is 
/  i  \a  v* 

*'=Us)    2^  = 

and  if  //is  the  total  available 
fall, 


=  remainder  of  fall  available  for  pressure-work. 


438  SLUICES. 

The  profile  AB  in  an  overfall  and  an  underflow  sluice 
should  coincide  with  the  parabolic  path  of  the  lowest  stream- 
lines  of  the  jet.  The  crest  of  the  overfall  should  be  properly 
curved,  and  the  inner  edges  of  the  underflow  opening  should 
be  carefully  rounded  so  as  to  eliminate  losses  due  to  contrac- 
tion. 

The  underflow  sluice-opening  should  also  be  normal  to  the 
axis  of  the  jet. 

Let  /IQ  be  the  head  above  the  crest  of  an  overfall  sluice. 
Then 


#j  being  the   width  of  the  crest,  and  c  the  coefficient  of  dis- 
charge.    The  width  b^  is  usually  3  or  4  ins.  less  than  the  width 
A  of  the  wheel. 
\  From  this  equation 


and  the  depth  of  water  over  the  crest  or  lip  is  usually  about 
$  ins. 

\   Again,  the  head  h^  (=  CD)  required  to  produce  the  velocity 
v^  at  the  point  of  entrance  B  is 


, 

102^' 

ib  per  cent  being  allowed  for  loss  due  to  friction. 
•     Thus   the   height   of  the   crest  A   above  B,   the   point   of 


entrance, 


=  AD  =  CD  -  CA  =  ^  - 


SLUICES.  439 

But  BA  is  a  parabola  with  its  vertex  at  A,  and  therefore, 
if  6  is  the  angle  between  the  horizontal  BD  and  the  tangent 
B  T  to  the  parabola  at  B, 


v?  sin2  6  _  i^v?        I 


30 


*v 

Also, 


The  head  available  for  pressure-work 


Let  a  be  the  angle  between  BT  and  the  tangent  to  the 
wheel's  periphery  at  B.      Then 

a  +  0  =  the  angle  EOF, 

BO  being   the   radius  to  the  centre   of  the  wheel   and   OFG9 
vertical. 

If  the  lowest  point  G'  of  the  wheel  just  clears  the  tail-race, 
the  head  available  for  pressure-work 

=  H  -  h^  =  FG'  =  OG'  -  OF 

ROT? 


=  r{(i  —  cos  EOF)  =  2rv  sin 

r^  being  the  radius  to  the  outer  periphery  of  the  wheel. 
If,  again,  the  water  enters  the  wheel  tangentially, 

a  —  O,  and  the  angle  EOF  —  6, 
so  that 

H—hi=  2rl  sin2-. 

If  the  sluice-opening  is  not  at  the  vertex  of  the  parabola, 
the  axis  of  the  opening  should  be  tangential  to  the  parabola. 


440 


BREAST   WHEELS. 


8.  Breast  Wheels. — These  wheels  are  usually  adopted  for 
falls  of  from  5  to  15  ft.,  and  for  a  delivery  of  from  5  to  80  cu. 
ft.  per  second. 

The  diameter  should  be  at  least  1 1  ft.  6  ins.,  and  rarely 
exceeds  24  ft.  The  velocity  («)  of  the  wheel's  periphery  is 
generally  from  3^  ft.  to  5  ft.  per  second,  the  most  useful 
average  velocity  being  about  4^  ft.  per  second. 

The  width  of  the  wheel  should  not  exceed  from  8  to  10  fL 

It  is  of  great  importance  to  retain  the  water  in  the  wheel  as 
long  as  possible,  and  this  is  effected  either  by  introducing  the 
water  at  the  inner  periphery,  Fig.  253,  or  by  surrounding  the 


FIG.  253. 


FIG.  254. 


water-arc  with  an  apron,  or  a  curb,  or  a  breast,  Fig.  254. 
which  may  be  constructed  of  timber,  iron,  or  stone.  In  this 
case  the  buckets  may  be  plane  floats,  as  the  curb  retains  the 
water,  but  they  should  be  set  at  an  angle  to  the  periphery  of 
the  wheel,  so  as  to  rise  out  of  the  water  with  the  least  resist- 
ance. 

Wheels  with  curbs  are  designated  as  high  breast,  breast, 
or  low  breast  according  as  the  water  reaches  the  wheel  near 
the  summit,  middle,  or  bottom,  while  if  there  is  no  curb  they 
are  termed  overshot,  middle-shot,  and  undershot,  respectively. 

The  depth  of  a  float  should  not  be  less  than   2.3  ft.,  and 


SPEED   OF  WHEEL.  441 

the  space  between  two  consecutive  floats  should  be  filled  to  at 
least  one  half,  and  even  to  two  thirds,  of  its  capacity.  The 
head  (measured  from  still  water)  over  the  sill  or  lip  should  be 
about  9  ins. 

The  play  between  the  outer  edge  of  the  floats  and  the  curb 
varies  from  \  in.,  in  the  best  constructed  wheels,  to  2  ins. 

The  distances  between  the  floats  is  from  i^  to  if  times  the 
head  over  the  sill  for  slow  wheels,  and  a  little  more  for  quick 
wheels. 

Breast  wheels  are  among  the  best  of  hydraulic  motors, 
having  an  efficiency  which  may  be  as  great  as  80  per  cent. 
The  efficiency  is  usually  about  70  per  cent  for  a  fall  of  about 
8  ft.,  and  50  per  cent  for  a  fall  of  4  ft. 

9.  Speed  of  Wheel. — The  water  leaves  the  buckets  and 
flows  away  in  the  race  with  a  velocity  not  sensibly  different 


FIG.  255. 

from  the  velocity  //  of  the  wheel,  which,  in  practice,  is  usually 
about  one  half  of  the  velocity  ^— J  with  which  the  water  enters. 

the  wheel. 

Let  b  be  the  width  of  the  wheel. 

Let  x  be  depth  of  the  water  in  the  lowest  bucket. 


442  MECHANICAL  EFFECT  OF  BREAST  WHEELS. 

Allowing  for  the  thickness  of  the  buckets,  the  play  between 
the  wheel  and  curb,  etc., 

Q  —  cbxu, 

c   being    an    empirical    coefficient    whose     average    value    is 
about  .9.      Hence 

10  Q 

«   =    --    7—. 

9  ox 

In  practice  b  is  often  taken  to  be  —  to  —  •.     It  is  important 

that  b  should  be  as  small  as  possible  and  hence  x  should  be 
3.3  large  as  possible,  its  value  being  usually  I  J  ft.  to  2  ft. 
-  !  'It1  must  be  borne  in  mind,  however,  that  any  increase  in 
the  value  of  x  will  cause  an  increase  in  the  weight  of  water 
lifted  by  the  buckets  as  they  emerge  from  the  race,  and  will 
therefore  tend  to  diminish  the  efficiency. 

10.  Mechanical  Effect.  —  Theoretically,  the  total  mechanical 
effect 


H  being  the  fall  from  the  surface  of  still  water  in  the  supply- 
channel  to  the  surface  of  the  water  in  the  tail-race. 
This,  however,  is  reduced  by  the  following  losses: 
(a)   Owing  to  frictional  resistance,  etc.,  there  is  a  loss  of 

11  2 
head  in  the  supply-channel  which  may  be  measured  by  v  —  —  , 

<5 

v  being  approximately  -£§  to  TT(T. 

The  head  required  to  produce  the  velocity  v^  at  entrance 


MECHANICAL   EFFECT  OF  BREAST   WHEELS.  443 

(b)  Let  of,  Fig.  256,  represent  in  direction  and  magnitude 
7;t  the  velocity  of  the  water 
entering  the  bucket. 

Let  ad,  in  the  direction 
of  the  tangent  to  the 
wheel's  periphery,  repre- 
sent the  velocity  u  of  the 
periphery  in  direction  and 
magnitude. 

Complete  the  parallelo-  FlG-  25&. 

gram  bd.  Then  ab  evidently  represents  the  velocity  V  of  the 
water  relatively  to  the  wheel.  This  velocity  V  is  rapidly 
destroyed,  the  corresponding  loss  of  head  being 

V*         1l2  4-  t',8  —  2111'    COS   V 


y  being  the  angle  daf. 

Assuming  that  the  water  enters  the  race  with  a  velocity 
equal  to  «,-  the  speed  of  the  wheel,  the  theoretical  useful  work 
per  pound  of  water  per  second  due  to  impact 

v?        V2         n>        u 
-  —  —  —  —  -  =  -4v,  cos  Y  —  u}, 

2/7"  2P"  2"'7~  fr^     *• 

i'  2  cos2  y       /* 
which  is  a  maximum  and  =  — —  =  — , 

when  7'L  cos  y  =  2u. 

In  practice  y  is  usually  about  30°,  and 

3?;i2 
the  maximum  useful  work  =  - 

Z2f' 

corresponding  to  the  relation  4^=  1/37', ,    or  //  =  .433^. 

To    diminish    as    much    as  possible    the  loss  in    shock    at 

V2 
entrance  due  to  the  dissipation  of  the  energy  —  in  eddy  motion, 

o 

the  direction  ab  of  the  relative  velocity  V  should  be  parallel  to 


444  MECHANICAL   EFFECT  OF  BREAST   WHEELS. 

the  arm  or  tangential  to  the  lip  of  the  bucket  and  should  there- 
fare  be  approximately  at  right  angles  to  the  wheel's  periphery. 

If,  at  the  point  of  entrance,  the  inlet  lip  is  the  lowest  point 
of  the  bucket,  the  water  flows  upwards,  and  the  relative  velocity 
V,  instead  of  being  wholly  destroyed  in  eddy  motion,  is  par- 
tially destroyed  by  gravity.  This  latter  is  again  restored  to 
the  water  on  its  return,  and  increases  the  wheel's  efficiency. 

For  a  given  speed  (#)  of  the  wheel,  the  velocity  (7^)  with 
which  the  water  should  reach  the  wheel  in  order  to  make  the 

y2 

loss  of  —    a  minimum  is  found  by  making  dV  —  o  in  eq.  (i), 

and  then 

0  =  7^.  dvl  —  u  cos  Y  -  dv\  > 
or  7't  =  u  cos  Y" 

This  is  an  impossible  relation,  as  it  makes  7^  <  ?/  and  the 
useful  work  negative.  In  fact  the  angle  afd  (=  baf )  in  such 
case  would  be  90°,  and  the  direction  af  of  vl  would  be  prac- 
tically tangential,  so  that  no  water  would  enter  the  wheel. 

Again,  for  a  given  velocity  7^  of  the  water  as  it  reaches  the 

V* 
wheel,  the  speed  of  wheel  which  would  make  the  loss  of  — 

2  2~ 

a  minimum  is  given  by 

o  •=.  u  .  du  —  7^  cos  Y  •  du, 
or  it  =  i\  c6s  Y' 

This  is  also  an  impossible  relation,  as  it  makes  the  useful 
work  nil. 

It  will  be  found  advantageous  to 
use  curved  or  polygonal  buckets  and 
not  plane  floats.  A  bucket,  for  ex- 


ample,  may  consist  of  three  straight 
portions,  ab,  be,  cd,  Fig.  257.  Of 
these  the  inner  portion,  cd,  should  be 
radial;  the  outer  portion,  ab,  is  nearly 
normal  to  the  periphery  of  the  wheel,  and  the  central  portion,, 
be,  may  make  angles  of  about  135°  with  ab  and  cd. 


MECHANICAL   EFFECT  OF  BREAST   WHEELS.  445 

Disregarding  all  other  losses,  the  theoretical  delivery  of 
the  wheel  in  foot-pounds 

=  WQ 

where    h2  =  total    fall  —  fall    (ht)    required    to    produce    the 
velocity  vr 

If  ?/  be  the  efficiency,  then,  according  to  the  results  of 
Morin's  experiments, 

77  =  .40  to  .45  if  /^  =  -H\ 

4 

V=  .42  to  .49if//1==|//; 
?t  =  .47  if  ^  ==  2-H\ 

?=.55  if //^l//. 

(V)  There  is  a  loss  of  head  due  to  frictional  resistance  along 
the  channel  in  which  the  wheel  works. 

Let  /  =  length  of  the  channel  (or  curb). 

Let  t  =  thickness  of  water-layer  leaving  the  wheel. 

Let  b  —  breadth  of  wheel. 

The  mean  velocity  of  flow  in  this  curb  channel  is  approx- 
imately -u,  and  the  loss  of  head  due  to  channel  friction 


f 


bt          2g         ~    3  bt       2' 

9 

where  /=  coefficiency  of  friction,  b  -\-  2t  =  wetted  perimeter, 
bt  =  water  area,  and  y  being  30°. 

(d)  There  is  a  loss  of  head  due  to  the  escape  of  water  over 
the  ends  and  sides  of  the  buckets. 

Let  s^  be  the  play  between  the  ends  of  the  buckets  and  the 
channel. 


446  MECHANICAL   EFFECT  OF  BREAST   WHEELS. 

Let  s.2  be  the  play  at  the  sides.      (sl  =  s2  ,  approximately.) 
Let    sl ,    z.z ,  .  .  .  zn    be  the  depths  of  water  in   a   bucket 
corresponding    to    n   successive    positions    in    its 
descent  from  the  receiving  to  the  lowest  point. 
Let  /j ,  /„,  .  .  .  /„  be  the  corresponding  water-arcs  meas- 
ured along  the  wheel's  periphery. 
The  orifice  of  discharge  at  end  of  a  bucket  =  bsr 
The  mean  amount  of  water  escaping  from  a  bucket  over  its 
end 


-f-  .  . 


c  being  the  coefficient  of  discharge. 

The  water  escapes  at  the  sides  as  over  a  series  of  weirs,, 
and  the  mean  amount  of  water  escaping  from  a  bucket  over  the 
sides 


Hence  the  total  loss  of  effect  from  escape  of  water 


per  sec.,  h  being  the  vertical  distance  between  the  point  of 
entrance  and  the  surface  of  the  water  in  the  tail-race 


(e)  There  is  a  loss  of  head  due  to  journal  friction. 

Let  W=  weight  of  wheel. 

Let  wl  =  weight  of  water  on  the  wheel. 

Let  rl    =  radius  of  wheel's  outer  periphery. 

Let  r'    —  radius  of  axle. 


MECHANICAL   EFFECT  OF  BREAST   WHEELS.  447 

Loss  per  second  of  mechanical  effect  due  to  journal  friction 


r  being  the  coefficient  of  journal  friction. 

There  is  a  loss  of  mechanial  effect  due  to  the  resistance  of 
the  air  to  the  motion  of  the  floats  (buckets),  but  this  is  prac- 
tically very  small,  and  may  be  disregarded  without  sensible 
error  A  deepening  of  the  tail-race  produces  a  further  loss  of 
effect,  and  should  only  be  adopted  when  back-water  is  feared. 

Hence  the  total  actual  mechanical  effect,  putting 


I  U\  I     V*         U2  -f  7.;  *  _    2UV    COS   y\ 

is  =  wQ  (H  —       —  wQ[v  -  -+- 

.V  2J        .      n     2^  2  I 


u(v,  cos  y-u) 


t 
WQ  - 


bt      3  2g  5  n 

wZ\l          -  v?\      wQ 

-  I  +  r^)+  —u(Vl  cos  Y- 

+1L  1SL,  -  X  W+  «,/«). 

^/        3  2g  ^ 


Hence    for    a    given    value    of  vv    the    mechanical    effect 
(omitting  the  last  term)  is  a  maximum  when 

v   cos  y 
u  =  -  —  --  (=  .433  X  ^  ,  if  y  =  30°). 

In  practice  the  speed  of  the  wheel  is  made  about  one  half 
of  the  velocity  with  whicri-  the  water  enters  the  wheel. 

For  a  given  speed  of  wheel,  and  disregarding  the  loss  of 


443 


EXAMPLE. 


u  cos  Y  =  O. 


effect  due  to  curb  friction,  which  is  always  small,  the  mechan- 
ical effect  is  a  maximum  for  a  value  of  i\  given  by 

.~wZ\\  -f  r       .    wQ 

n   i     g     tl 
or 

u  cos  Y 

*!  = 


The  loss  by  escape  of  water,  viz.,  c^2g  ,  varies,  on  an 
average,  from  10  to  15  per  cent  of  the  whole  supply,  so  that 

c  */2g—  varies  from  —  to  —  . 
*  n  10        20 

Ex.  The  buckets  of  a  low  breast  wheel,  of  24  ft.  diameter,  are  half 
filled  with  water  which  flows  from  a  flume  through  a  vertical  rectangular 
sluice-opening  at  the  rate  of  15  cu.  ft.  per  second.  The  linear  speed  of 
the  wheel's  periphery  is  5  ft,  per  second.  At  the  point  of  admission 
the  inflowing  jet  has  a  velocity  of  10  ft.  per  second  and  makes  an  angle  of 


FIG.  258. 


30°  with  the  rim.  The  total  available  fall  is  8|  ft.  Find  (a)  the  position 
of  the  point  of  admission;  (&}  the  work  done  by  impact  and  weight; 
{*)  the  position  and  dimensions  of  the  sluice-opening,  the  depth  of  the 
shrouding  being  12  ins. 


S4GEBIEN   WHEEL  449 

(a)  Let  OB  be  the  radius  to  the  point  of  admission  JB,  and  let  <f>  be 
its  inclination  to  the  vertical. 

Draw  the  vertical  OG  and  the  horizontal  BF. 

Theoretically,  hi ,  the  head  required  to  develop  a  velocity  of  10  ft.  per 
second, 

=£=".*• 

Then  8|  ~  i^  =  6|f  ft.  =  head  available  for  work  by  weight 
=  the  vertical  fall  on  the  wheel 

=  FG. 

Therefore      cos  0  =  -^  =  -'  ~    T*  =  .421875, 

and  (p  =  65°  3',  defining  the  position  of  B. 

(&)  The  useful  theoretical  work  done  by  impact 

=  -^'5 -5(io  cos  30°  -  5)=  536.i33ft.-lbs. 

The  useful  theoretical  work  done  by  weight 

=  62^  .  15  .  6\l  =  6503.906  ft.-lbs., 
and  the  combined  useful  work  =  7040.039  ft.-lbs. 

(c)  Let  AD,  BD  be  the  vertical  and  horizontal  distances  of  the  lift 
A  from  B. 

The  angle  between  the  direction  of  z/i  at  B  and  the  horizontal 
==  0  -  30°  =  35°  3' 

Therefore  AD  —  ^—  sin2  35°  3'  =  .51533  ft. 

04 

and  BD  =  7—  sin  70°  6'  =  1.4692  ft. 

Again,  the  width  of  the  wheel  =  j —      —  =  6  ft., 

and  the  width  of  the  sluice  may  be  taken  to  be  about  3  ins.  less 
than  this,  or  5!  ft.  The  head  over  the  lip  =  IT9^  — •5I533  —  1.0472; 
the  average  velocity  of  flow  through  the  sluice  =  .  9  4/64  x  1.0472  = 

7.3656  ft.  per  second,  and  the  depth  of  the  sluice-opening  =  — — 
=  -354  ft. 

II.  Sagebien  Wheels,  Fig.  259,  have  plane  floats  inclined 
to  the  radius  at  from  40°  to  45°  in  the  direction  of  the  wheel's 
rotation.  The  floats  are  near  together  and  sink  slowly  into 
the  fluid  mass.  The  level  of  the  water  in  the  float-passages 
gradually  varies,  and  the  volume  discharged  in  a  given  time 
may  be  very  greatly  changed.  The  efficiency  of  these  wheels 


45° 


OVERSHOT   WHEEL. 


is  over  80  per  cent,  and  has  reached  even  90  per  cent.      The 
action  is  almost  the  same  as  if  the  water  were  transferred  from 


FIG.  259. 

the  upper  to  the  lower  race,  without  agitation,  frictional  resis- 
tance, etc.,  flowing  away  without  obstruction  into  the  tail-race. 

12.  Overshot  Wheels. — Since  the  introduction  and  develop- 
ment of  the  turbine  these  wheels  have  become  almost  obsolete. 
They  have  been  considered  among  the  best  of  hydraulic  motors 
for  falls  of  8  to  70  ft.  and  for  a  delivery  of  3  to   25  cu.  ft.  per 
second,   and  have  proved  especially  useful  for  falls  of  12  to 
20  ft.      The  efficiency  of  overshot  wheels  of  the  best  construc- 
tion is  from  .70  to  .85. 

The  thickness  of  the  sheet  of  water  passing  through  the 
sluice  on  to  the  wheel  rarely  exceeds  4  or  5  ins.,  and  is  often 
less  than  2  ins. 

If  the  level  of  the  head-water  is  liable  to  a  greater  variation 
than  2  ft.,  it  is  most  advantageous  to  employ  a  pitch-back  or 
high  breast  wheel,  which  receives  the  water  on  the  same  side 
as  the  channel  of  approach. 

13.  Wheel  Velocity. — This  evidently  depends    upon    the 
work  to  be  done,  and  upon  the  velocity  with  which  the  water 
arrives  on  the  wheel.      Overshot  wheels  should  have  a  low  cir- 


OVERSHOT   WHEEL. 


45* 


cumferential  speed,  varying  from  10  ft.  per  second  for  large 
wheels  to  3  ft.  per  second  for  small  wheels,  and  should  not  be 
less  than  2|-  ft.  per  second.  At  a  higher  speed  than  6  ft.  per 
second,  if  the  buckets  are  more  than  two  thirds  full,  the 
efficiency  does  not  exceed  60  per  cent. 

In  order  that  the  water  may  enter  the  buckets  easily,  its 
velocity  should  be  greater  than  the  peripheral  velocity  of  the 
wheel. 

14.  Effect  of  Centrifugal  Force. — Consider  a  molecule  of 
weight  w  in  the  * '  unknown  ' '  surface  of  the  water  in  a  bucket 
(Fig.  260).      At  each  moment  there 
is  a  dynamical  equilibrium  between 
the  "forces  "  acting  on  ;;/,  viz.  :  (i) 
its    weight    w ; 


(2)    the    centrifugal 
the  resultant    T  of 
the  neighboring  reactions. 

Take    MF  =  w.     MG  =  ™ 


force    -<&r\   (3) 

o 


and  complete  the  parallelogram  FG. 
Then  J///  =  T.  The  direction  of 
T  is,  of  course,  normal  to  the  surface 
of  the  water  in  the  bucket. 

Let  HM  produced  meet  the  ver- 
tical through  the  axis  O  of  the  wheel 
in  E.  Then 


MG 


—  G&r 

_  g 


FH       OM 


MF  "       w 
and  therefore 


MF 


OE' 


FIG.  260. 


taking  g  —  32  ft.  and  n  being  the  number  of  revolutions  per 
minute. 


45  2  OVERSHOT   WHEEL. 

Thus  the  position  of  E  is  independent  of  r  and  of  the  posi- 
tion of  the  bucket,  so  that  all  the  normals  to  the  water-surface 
in  a  bucket  meet  in  E,  and  the  surface  is  the  arc  of  a  circle 
having  its  centre  at  E,  or,  rather,  a  cylindrical  surface  with 
axis  through  E  parallel  to  the  axis  of  rotation. 

15.  Weight  of  Water  on  Wheel  and  Arc  of  Discharge.- 
Let  Q  —  volume  supplied  per  sec.  ,  and  N  =  number  of  buckets. 

Then  -  -  =  number  of  buckets  fed  per  second, 

271 

and    —jrf-  =  volume  of  water  received  by  each  bucket  per  sec. 
NGO 

Hence  the  area  occupied  by  the  water  until  spilling  com- 
mences =  .  ,T  ,  b  being  the  bucket's  width  (=  width  of  wheel 


between  the  shroudings). 

The  water  flows  on  to  the  wheel  through  a  channel  (Fig. 
261),  usually  of  the  same  width  b  as  the  wheel,  and  the  supply 
is  regulated  by  means  of  an  adjustable  sluice,  which  may  be 
either  vertical,  inclined,  or  horizontal. 

When  the  water  springs  clear  from  the  sluice,  as  in  Fig. 
261,  the  axis  of  the  sluice  should  be  tangential  to  the  axis  of 
the  jet,  and  the  inner  edges  of  the  sluice-opening  should  be 
rounded  so  as  to  eliminate  contraction. 

Let  j/,  z  be  the  horizontal  and  vertical  distances  between 
the  sluice  and  the  point  of  entrance. 

Let  T  be  the  time  of  flow  between  the  sluice  and  entrance. 

Let  vQ,-vl  be  the  velocities  of  flow  on  leaving  the  sluice 
and  on  entering  the  bucket. 

Then 

vl  cos  (y 
z^sin  (r 
and 


6  being  the  angular  deviation  of  the  point  of  entrance  from  the 


OVERSHOT  WHEEL. 


45$ 


summit,  and  y  the  angle  between  the  direction  of  motion  of 
the  water  and  the  wheel  at  the  point  of  entrance. 

If  the  bed  of  the  channel  is  horizontal,  and  if  also  the  sluice 
is  vertical,  opening  upwards  from  the  bed,  and  is  of  the  same 
width  b  as  the  wheel,  then 


FIG.  261. 

/  being  the  depth  of  sluice-opening  and  h^  the  effective  head 
over  the  sluice.  This  effective  head  is  about  T97  of  the 
actual  head. 


32,  -=•  = 


gives  the  delivery  per  foot 


Thus,  taking  g 

width  of  wheel. 

Taking  .6  ft.  and  3.6  ft.   as  the  extreme  limits  between 
which  /Zj  should  lie,  and  .2  ft.  and  .33  ft.  as  the  extreme  limits 


454  OVERSHOT   WHEEL. 

between  which  /  should  lie,  then  -7  must  lie  between  the  limits 

o 

1.24  and  5,  and  an  average  value  of  ->-  is  3.      Thus  the  width 

of  the  wheel  should  be  on  the  average  _   —  . 

Again,  disregarding  the  thickness  of  the  buckets,  the 
capacity  of  the  portion  of  the  wheel  passing  in  front  of  the 
water-supply  per  second 

r  a      ir  _  rf'Y  )  I         d\ 

-  —  V  =  bd(*)\r^  --  1  =  bdr^c*),  approximately, 


t 

rl  being  the  radius  and  u  the  velocity  of  the  outer  circumference 
of  the  wheel,  d  the  depth  of  the  shrouding,  or  crown,  and  n 
the  number  of  revolutions  per  minute. 

Only  a  portion,  however,  of  the  space  can  be  occupied  by 
the  water,  so  that  the  capacity  of  a  bucket  is  mubd,  m  being 
a  fraction  less  than  unity  and  usually  £  or  J.  For  very  high 
wheels  m  may  be  -J-.  Hence 

mbdu  =  Q. 

Therefore  mdu  =*-r. 

o 

The    delivery  (-7-)   per  foot  of  width  must   not   exceed    a 

certain  limit,  otherwise  either  d  or  u  will  be  too  great.  In  the 
former  case  the  water  would  acquire  too  great  a  velocity  on 
entering  the  buckets,  which  would  lead  to  an  excessive  loss 
in  eddy  motion  and  a  corresponding  loss  of  efficiency  ;  while 
if  the  speed  u  of  the  wheel  is  too  great,  the  efficiency  is  again 
diminished  and  might  fall  even  below  40  per  cent. 

The  depth  of  a  bucket  or  of  the  shrouding  varies  from  10 
to  1  6  ins.,  being  usually  from  10  to  12  ins.,  and  the  buckets 


OVERSHOT   WHEEL. 


455 


are  spread  along  the  outer  circumference  at  intervals  of  12  to 
14  ins.  The  number  of  the  buckets  is  approximately  5^  or 
6rl ,  rl  being  in  feet. 

The  efficiency  of  the  wheel  necessarily  increases  with  the 
number  of  the  buckets,  but  the  number  is  limited  by  certain 
considerations,  viz.  :  (a)  the  bucket  thickness  must  not  take 
up  too  much  of  the  wheel's  periphery;  (b)  the  number  of  the 


o  ;— 


FIG.  262. 


buckets  must  not  be  so  great  as  to  obstruct  the  free  entrance 
of  the  water ;.  (c)  the  form  of  the  bucket  essentially  affects  the 
number. 

Let  the  bucket,  Fig.  262,  consist  of  two  portions,  an  inner 


456  OVERSHOT  WHEEL. 

portion  bcy  which  is  radial,  and  an  outer  portion,  cd\  c  being  a 
point  on  what  is  called  the  division  circle.  The  length  be  is 
usually  one  half  or  two  thirds  of  the  depth  d  of  the  shrouding. 

Take  be  =  \d. 

It  may  also  be  assumed  without  much  error  that  the  water- 
surface  ad  is  approximately  perpendicular  to  the  line  cd,  so 
that  the  angle  eda  is  approximately  a  right  angle. 

The  spilling  evidently  commences  when  the  cylindrical 
surface,  having  its  axis  at  e  and  cutting  off  from  the  bucket  a 

water-area  equal  to  ,  passes  through  the  outer  edge  d  of 

the  bucket. 

Let  ft  be  the  bucket  angle  cOd. 

Let  0  be  the  inclination  of  Od  to  the  horizon. 

Let  0  be  the  inclination  of  ad  to  the  horizon. 

Let  rl  be  the  radius  of  the  outer  periphery. 

Let  R  be  the  radius  of  the  division  circle. 

Let  r2  be  the  radius  of  the  inner  periphery. 

Then 

_g_      J0e  _  cos  (0  ±  0) 

r^       Od  ~         sin  0 

the  sign  being  plus  or  minus  according  as  the  bucket  is  below 
or  above  the  horizontal,  and  in  the  latter  case,  if  6  =  0,  then 
/•jCo2  =  g  sin  0. 

Again, 

af  =  fd  tan  (0  -f-  0),  approximately. 

Therefore 

the  area  dfa  —  —  tan  (6  +  0)  =  •  -  tan  (V  +  0), 

where  d  =  rl  —  ry      Hence 

the  area  abed  =  area  cod  —  area  bof  —  arpa  dfa 


OYERSHOT  WHEEL.  457 

Equations  (i)  and  (2)  give  6  and  0,  and  therefore  the  posi- 
tion of  the  bucket  when  spilling  commences. 

The  bucket  will  be  completely  emptied  when  it  has  reached 
a  position  in  which  cd  is  perpendicular  to  a  line  from  e  to 
middle  point  of  cd,  or,  approximately,  when  edc  is  a  right 
angle. 

Let  6l  ,  0j  be  the  corresponding  values  of  6  and  0,  and  let 
yl  be  the  angle  between  cd  and  the  tangent  at  d  to  the  wheel's 
periphery.  Then 


and 


=  9o°  - 


sn 


two  equations  giving  0L  and  #r 

Also,  if  ck  is  drawn  perpendicular  to  <?*/, 

dk      r.  —  R  cos  /?      r. 
tan  Yl  =  cot  cdk  =  c-g  =  -1  R  sin  ft     =  £  cosec  /?  -  cot  /?. 

The  vertical  distance  between  the  points  where  spilling 
begins  and  ends,  viz.,  ^(sin  Ol  —  sin  (?)  can  now  be  deter- 
mined. 

The  pitch-angle  (=  ifj)  is  the  angle  between  two  consecu- 

360° 
tive   buckets  so  that  ib  =  —  r^-.      In  order  to  obtain  a  small 

N 

angle  (==  y})  between  the  lip  of  the  bucket  and  the  wheel's 
periphery,  it  is  usual  to  make  the  bucket  angle  ft  greater 
than  rp. 

For  example, 

5.       536o0      450° 


-  4r   "  4   »  W  ' 

The  interval  between  the  buckets  should  be  at  least  suffi- 
cient to  prevent  any  bucket  dipping  into  the  one  below  at  the 
moment  the  latter  begins  to  spill. 


458 


OVERSHOT   V/HEEL. 


Let  coc1 ',  Fig.  263,  be  the  division  angle,  and  t  the  thick- 
ness of  the  bucket. 


Then 


FIG.  263. 


'==      tan  (0  +*)  =      tan  0, 


approximately,  and  therefore 

J$(r2/3  + 1  -  -  tan  6)  =  mrr       ...     (3) 

Also,  by  equation  (2), 

rP  r2  H2  2;rQ 

'  +  Mfi-       '     '     (4) 


These  last  two  equations  give  N  and  6. 
The  number  of  buckets  may  also  be  approximately  found 
from  the  formula 


15.  Form   and   Capacity   of   Bucket.  —  In   practice   the 
bucket  may  be  delineated  as  follows  : 

In  Fig.  263  let  dd'  =  distance  between  two  buckets. 


CAPACITY  OF  BUCKET.  459 

Take  dd"  =  -dd'  to—dd'}  also  take  be  =  — ,  and  join  dc. 

This  gives  the  form  of  a  suitable  wooden  bucket. 

If  the  bucket  is  of  iron,  circular  arc  is  substituted  for  the 
portions  be,  cd. 

Again,  let  pm,  Fig.  264,  be  the  thickness  of  the  stream 
just  before  entering  the  bucket. 


Let  dn  be  the  thickness  of  the  stream  just  after  entering 
the  bucket. 

Let  yl  be  the  angle  between  the  bucket's  lip  and  the  wheel's 
periphery. 

Then 

mbduv  =  capacity  of  bucket  =  bv^  .  pm  =  bV .  dn 

=  bv^dp  sin  y  =  b  V .  dp  .  sin  y\ » 

and  therefore 

mduv  mdu^ 

P  '  ~  vl  sin  Y  "  V  sin  y^ 

Now  overshot  wheels  cannot  be  ventilated,  and  it  is  con- 
sequently necessary  to  leave  ample  space  above  the  entering 


460  CAPACITY  OF  BUCKET. 

stream  for  the  free  exit  of  air.      Thus,  neglecting  float  thick- 
ness, 

O 

=  the  distance  between  consecutive  floats 


N 

=  dd'  (Fig.  263)  >  dp  > 

and  N,  the  number  of  buckets, 

^Fsin  yl 


For  efficient  action  the  number  of  the  buckets  is  much  less 
than  the  limit  given  by  this  relation,  often  not  exceeding  one 
half  of  such  limit. 

If  y\  is  very  small,  V  =  v^  —  ul ,  approximately,  and  there- 
fore 


md 


^T      —  .  sin  v,  /v.          \ 
N<-  ig-,). 


The  capacity  of  a  bucket  depends  upon  its  form ;  and  the 
bucket  must  be  so  designed  that  the  water  can  enter  freely  and 
without  shock,  is  retained  to  the  lowest  possible  point,  and  is 
finally  discharged  without  let  vor  hindrance.  Hence  flat 
buckets,  Fig.  265,  are  not  so  efficient  as  the  curved  iron 
bucket  in  Fig.  268  and  as  the  compound  bucket  made  of  three 
or  two  pieces  in  Figs.  266,  267,  and  269.  The  resistance  to 
entrance  is  least  in  the  curved  bucket,  as  there  are  no  abrupt 
changes  of  direction  due  to  angles.  The  capacity  of  a  com- 
pound bucket  may  be  increased,  without  diminishing  the  ease 
of  entrance,  by  making  the  inner  portion  strike  the  inner 
periphery  at  an  acute  angle,  Fig.  269.  The  objection  to  this 
construction,  especially  if  the  relative  velocity  V  is  large,  is 
that  the  water  tends  to  return  in  the  opposite  direction  and 
escape  from  the  bucket. 


FIG.  265. 


CAPACITY  OF  BUCKET. 
FIG.  266. 


461 


FIG.  267. 


/ 


FIG.  268. 


FIG.  269. 


FIG.  270. 


462  CAPACITY  OF  BUCKET. 

Let  bed,  efg,  Fig.  270,  represent  two  consecutive  buckets 
of  an  overshot  wheel  turning-  in  the  direction  shown  by  the 
arrow. 

Water  will  cease  to  enter  the  bucket-space  between  bed 
and  efg,  and  impact  will  therefore  cease,  when  the  upper 
parabolic  boundary  of  the  supply-stream  intersects  the  edge  d. 
The  last  fluid  elements  will  then  strike  the  water  already  in 
the  bucket  at  a  point  M,  whose  vertical  distance  below  d  may 
be  designated  by  z.  The  velocity  i\'  with  which  the  entering 
particles  reach  M  is  given  by  the  equation 


v{-_-_    \V*+2gZ  ......        (I) 

Again,  while  the  fluid  particles  move  from  d  to  J/let  the 
buckets  move  into  the  positions  d'c'b1  ',  e'f'g'. 
Let  arc  dd'  =  s^  =  ee'  . 
Let  arc  dM  =  s2. 

Let  T  be  the  time  of  movement  from  d  to  d'  (or  d  to  M}. 
Then 


sl  =  uT 


and 


assuming  that  the  mean  velocity  from  d  to  M  is  an  arithmetic 
mean  between  the  initial  and  final  velocity  of  entrance.      Thus 


Also,  since  the  angle  between  */J/and  the  wheel's  periphery 
is  small,  it  may  be  assumed  that 

the  arc  dM  =  de  -\-  ef-\-  ee'  ,  approximately, 

^i  ~  ^ 

~"     ^ 


y          tv,  —  u       27rr,    v.  —  u  \ 

NOTE.  —  ef—  ed—  —  ed^     -  —  -^  .  -     —  ,  nearly.  ] 

u  u  N         u  y  ) 


EXAMPLES.  463 

Thus 


and  by  equations  (2)  and  (3), 

+  7'/  ~  2* 


2«       r    N  u' 

an  equation  giving  approximately  the  distance  ^  passed 
through  by  a  float  during  impact.  The  buckets  can  now  be 
plotted  in  the  positions  they  occupy  at  the  end  of  the  impact. 
The  amount  of  water  in  each  bucket  being  also  known,  the 
water-surface  can  be  delineated,  and  hence  the  vertical  distance 
z  can  be  at  once  found. 

Ex.  i.  Find  the  angular  depression  of  the  water-surface  below  the 
horizontal  (a)  when  the  bucket  lip  is  37°  14'  above  the  centre,  and  (b) 
when  the  bucket  lip  is  on  a  level  with  the  centre;  also  find  (c)  the 
position  of  the  bucket  below  the  centre  when  a  horizontal  through  the 
lip  bisects  the  angle  between  the  water-surface  and  the  radius  to  the  lip. 
The  wheel  has  a  diameter  of  32  ft.  and  makes  io£  revolutions  per  minute. 

The  angular  velocity  (»  =  —  ~-r—  =  — .     Then 

7    60        10 

-~,  . .  .  sin  0  121 

O) 


32\io/  ~~  cos  (37°  14'  —  0)  ~~  200' 

„,        t  200       cos  (37°  14'—  0) 

Therefore      —  =  =  cos  37    H'  cot  $  +  sin  37°  14', 


121 

or  cot  0  =  1.316     and     0  =  57°  14'. 

121  sin  0 

(0)  • —  =  —  — —  =  tan  0  =  .60?, 

200       cos  (0+0) 

and  0  =  31°  10'. 

121  sin   0  sin  0  sin  0 


to 


200       cos  (6  4-  0)       cos  20       i—2  sina  0' 


IOO  I 

or  sin    0  +  sin  0  =  — , 

121  2 

and  sin  0  =  .4058, 

or  0  =  23°  56'  =  6. 


464 


EXAMPLES. 


e 


Ex.  2.  An  overshot  wheel  has  a  diameter  of  32  ft.,  a  12-in.  crown, 
and  its  peripheral  speed  is  4  ft.  per  second.     The  lip  of  the  bucket  is  j£ 

ins.  thick.  Water  enters  the 
wheel  in  a  direction  inclined  at 
60°  to  the  vertical  at  a  point  12° 
30'  from  the  summit  and  with  a 
velocity  of  16  ft.  per  second. 
Spilling  commences  at  120°  from 
the  summit.  Find  (a)  the  relative 
velocity  ( J7)  at  admission  ;  (<$)  the 
angle  between  the  horizontal  and 
the  water-surface  at  i°  47'  33'' 
above  and  at  30°  below  the  cen- 
tre; (c)  the  angle  (y<)  between  the 
bucket  lip  and  the  nm  :  (d}  the 
point  where  the  bucket  is  emp- 
tied ;  (e)  the  bucket  angle;  (/)  the 
elbow  angle  ;  (g]  the  number  of 
buckets ;  (h)  the  bucket  water 
area. 

At  the  point  of  admission  d 
let  dgh  be  the  triangle  of  veloci- 
ties so  that  dg  —  1 6  ft.,  dh  =  4  ft., 
and  the  angle  gdh  =  30°  —  12°  30' 
=  17°  30'. 

Assuming  also  that  the  water 
enters  without  shock,  the  relative 
velocity    V  —  (dg)    is    parallel  to 
the  bucket  arm  cdt  and  the  angle 
^cdk—  ^3  —  angle  ghk. 

Then 


FIG.  270. 


(a)  Fa  =  y^2  =  dg*  +  dJi*  —  ?.  .  gd .  dh  cos  17°  30' 

=  i62  +  42  —  2  .  16  . 4  cos  17°  30' 
=  149.9242, 
and  V  =  12.2443  ft-  Per  sec- 

(£)  When  0  =  i°  47'  33", 

32      =  cos  (i°  47'  33"  -  0)  = 
i6(T\)2  sin  0 

or  cot  0  ='  32  sec  i°  47'  33"  -  tan  i°  47'  33"» 

and  0=i°  47'  33". 


When  B  =  30°, 


cos  (30°  +  0) 
sin  0 


=«  32, 


EXAMPLES. 

or  cot  0  =  32  sec  30°  +  tan  30°  =  37.5277» 

and  0  =  i°  31'  24". 

sin  17°  30'  _  12.2443 
sin  yi  10 

12.2443 
or  cosec  yi  —  ^~  cosec  17    30 '  =  2.545, 

and  yl  =  23°  8'. 

(d)  At  the  point  //s,  where  the  spilling  is  completed,  Odzk9  is  a  right 
angle  and  the  angle  Od*e  =  angle  c*d*k*  =  y\     Then 

sin  0i      ed*       r\        i6/4\2       i 
sin  yi  ~  oe  ~~  g  ~~  32  \i6/  ~  32' 


sin  23°  8' 

or  sin  0i  =  —  —  =  .0122772, 

32 

and  0i  =  o°  42'. 

Therefore  GI  =  90°  —  (7,  +  0,)  =  66°  10', 

and  the  bucket  is  emptied  at  90°  +  66°  10'  =  156°  10'  from  the  summit, 
(f)  n  =  1 6';     A3  =  1 5V.     Therefore 

tan  23°  8'  =  tan  yi  = •—-  — =  .4272, 

This  last  equation  is  easily  reduced  to  the  form 

cos2  ft  —  1.7458  cos  fj  =  —  .74674, 
and  cos  ft  —  .9962, 

or  ft  =  5°  8', 

121    . 

=  -   —  in  circular  measure. 
1350 

(/)  The  elbow  angle  Ocd  —  180°  —  ft  —  Ode  =  18°  -  ft  —  (90°  —  ^i> 

=  90°  -  5°  8'  +  23°  8" 
=  108°. 

44 


or  N  =  79.8,  say  80. 

An  empirical  approximate  rule  makes 

44     16 


466  EXAMPLES. 

(h)   -  —  sin  5°  8'  -  =     tan  30  +  water  area  of  bucket* 

Therefore 

the  water  area  =  11.09475  —  10.08333  ~"  .28867 
=       .72275  sq.  ft. 
=  104  sq.  ins. 

Ex.  3.  An  overshot  water-wheel,  of  40  ft.  diameter,  is  12  in.  wide  and 
lias  a  9.6-in.  shrouding.  The  pitch-angle  is  4°  and  the  thickness  of  the 
bucket  lip  is  i  in.  At  the  point  where  spilling  commences  the  bucket 
water  area  is  24-^  sq.  ins.  Find  the  number  of  buckets,  the  point  where 
spilling  commences,  and  the  angle  between  the  rim  and  the  bucket  lip. 

r,  =  20  ft.;     7?  =  20  —  -  —  -  ;  =  19.6  ft.;     r-i  —  20  —  .8  =  19.2  ft. 
Take  /3,  the  bucket-angle,  =  --4°  =  5°. 

s  I  .8  TO.  2 

Then  19  .  2   x  7f~  +        —  —  tan  0  =  27t-^—, 

i  80       12        2  A 

20    X     19.6     .  (IQ.2)2        5  (.8)3»  24^ 

and  -  sin  5°  -  —  ie-%-  =  ------  tan  6  +  —  -. 

2  2        1  80         2  144 

Hence  N  —  164.5, 

and  tan  9  —  2.56,     or     0  =  68°  40'. 

The  empirical  formula  gives 

2itfi       44  20 


Again,  tan  yi  =  -  —  >  cosec  5°  —  cot  5°  =  .27782, 

and  yi  =  i$°  32'. 

Ex.  4.  One  foitrtJi  of  the  theoretic  capacity  of  a  bucket  is  filled  with 
water.  The  angle  between  the  bucket  lip  and  the  wheel's  periphery  is 
20°,  the  radius  to  the  outer  periphery  is  18  ft.  and  the  depth  of  the  crown 
is  12  ins.  If  the  velocity  of  the  water  at  entrance  is  twice  that  of  the 
•wheel's  periphery,  find  the  greatest  number  of  buckets  theoretically 
possible. 

The  number 


<  103.2. 
The  actual  number  may  be  about  two  thirds  of  this,  or  69. 


MECHANICAL   EFFECT  OF  OVERSHOT  WHEELS. 


467 


16.  Useful  Effect.— (a)  Effect  of  Weight.— 1\\z  wheel 
should  hang  freely,  or  just  clear  the  tail-water  surface,  and  the 
total  fall  is  measured  from  the  surface  of  the  water  in  the  tail- 
race  to  the  water-surface  just  in  front  of  the  sluices  through 
which  the  water  is  brought  on  to  the  wheel. 

Let  hl ,    Fig.   272,   be   the  vertical   distance   between   the 


FIG.  272. 

centres  of  gravity  of  the  water-areas  of  the  first  and  last  buckets 
before  spilling  commences.      Then 

hj  —  R  cos  6  +  rl  sin  6,  very  nearly. 

Let  //2  be  the  vertical  distance  between  the  centres  of 
gravity  of  the  water-area  of  the  bucket  which  first  begins  to 
-spill  and  the  point  at  which  the  spilling  is  completed.  Then 

h2  =  r^sin  0t  —  sin  0),  very  nearly. 

The  useful  work  per  sec.  =  wQ(ht  -j-  kh2),  k  being  a  frac- 
tion <  I  and  approximately  =  .5. 

Let  AQ  be  the  water-area  in  the  bucket  which  first  begins 
to  spill. 


468  MECHANICAL  EFFECT  OF  OVERSHOT   WHEELS. 

Between  this  bucket  and  the  one  which  is  first  emptied,, 
i.e.,  in  the  vertical  distance  k2  ,  insert  s  buckets,  at  equal  dis- 
tances apart,  and  let  their  water-areas  AI}  A2,  A3,  .  .  .  As  be 
carefully  calculated. 

Let  Qm  be  the  mean  amount  of  water  per  bucket  in  the 
discharging  arc. 

Let  Am  be  the  mean  water-area  per  bucket  in  the  discharg- 
ing arc. 

Then 

A9  +  Al  +  A,+  ..  .+A,_t  +  As 
A<- 


The  value  of  £  can  now  be  easily  .found,  since 

k  -  °-  -  ^ 

'  Q  "  At- 

Let  q  be  the  varying  amount  of  water  in  a  bucket  from 
which  spilling  is  taking  place,  and  at  any  moment  let  y  be 
the  vertical  distance  between  the  outer  edge  of  the  bucket  and 
the  surface  of  the  water  in  the  tail-race. 

q  is  a  function  of  y  and  depends  upon  the  contour  of  the 
water  in  the  bucket- 

Let  Y  be  the  mean  value  of  ^  y  between  the  points  where 
spilling  begins  and  ends,  i.e.,  for  values  r{  and  y.,  of  y.  Then 


since 

r  / 

q  .  dy. 


r  r 

I  y  .  dq  =  yq  -  -     I  q 

«./  is 


Again,  the  elementary  quantity  of  water,  dq,  having  an 
initial  velocity  equal  to  that  of  the  wheel,  viz.,  u,  falls  a  dis- 
tance y  and  acquires  a  velocity  =  Vuz  -f-  2gy. 


MECHANICAL   EFFECT  OF  OVERSHOT   WHEELS.  .469 

Thus    it   flows    away   in    the    tail-race,  causing    a   loss   of 

w  .  dq 
energy  =  ~y-(^2  +  2£T)  =  ™  - 

Hence  the   total  loss  of  energy  between  the  points  where 
spilling  begins  and  ends 


Overshot  and  pitch-back  wheels  do  not  work  well  in  back- 
water, as  they  lift  a  greater  or  less  weight  of  water  in  rising 
above  the  surface. 

If  the  water-level  in  the  race  is  liable  to  variation  it  is 
better  to  diminish  the  diameter  of  the  wheel  and  design  it  so 
that  it  may  never  be  immersed  to  a  greater  depth  than  12  ins. 

(b)  Effect  of  Impact.  —  The  head  //'  required  to  produce  the 
velocity  v^  with  which  the  water  reaches  the  wheel  is  theoreti- 


cally    l    ;   but  as  there  is  a  loss   of  at  least   5  per  cent  in  the 

7/2 

most  perfect  delivery,  it  is  usual  to  take  //'  =  v— — ,  an  average 

value  of  v  being  I .  I . 

Let  the  water  enter  the  bucket  in  the  direction  ac,  Fig. 
273.  Take  ac  =  i\.  The  water  now  moves  round  with  a 
velocity  u  (assumed  the  same  as  that  of  the  division  circle), 
and  leaves  the  wheel  with  the  same  velocity.  Take  ab  in  the 
direction  of  the  tangent  to  the  division  circle  at  the  point  of 
entrance  =  u.  The  component  be  represents  the  relative 
velocity  V  of  the  water  with  respect  to  the  bucket,  and  this 
velocity  is  wholly  destroyed,  ab  must  necessarily  be  parallel 
to  the  outer  arm  of  the'  bucket,  so  that  there  may  be  no  loss, 
of  shock  at  entrance.  Then  the  impulsive  effect 

'  __  —  -  ' 
2 


But 


MECHANICAL   EFFECT  OF  OYERSHOT   WHEELS. 


^u  cos  y 


-y  being  the  angle  through  which  the  water  is  deviated  from 
its  original  direction  at  the  point  of  entrance. 


.  273. 


Hence  the  impulsive  effect 


wQ 
=  —  ruto  cos  y  -  u), 

o 


and  the  TOTAL    USEFUL    EFFECT 

wO 


^u(v1cos  y  —  u)  —  loss  due  to  journal  friction. 

o 


Designating  the  first  two  terms  of  this  expression  by  P, 
the  loss  due  to  journal  friction 


=    JLll—+    W\-U, 
(   U  )  ^l 

p  being  the  radius  of  the  axle,  and  ^Fthe  weight  of  the  wheel. 


EXAMPLE.  47* 

Ex.  An  overshot  wheel  weighing  20,000  Ibs.,  with  a  12-in.  crown  and 
of  40  ft.  diameter,  receives  400  cu.  ft.  of  water  per  minute  and  revolves 
in  6-in.  bearings.  The  water  enters  the  buckets  at  12°  from  the  wheel's 
summit,  with  a  velocity  of  16  ft.  per  second  and  at  an  angle  of  10°  with 
the  wheel's  periphery,  which  moves  with  a  linear  velocity  of  9  ft.  pet- 
second.  Spilling  commences  and  is  completed  at  points  which  are 
respectively  140°  and  160°  from  the  wheel's  summit.  Determine  the 
power  of  the  wheel  and  its  efficiency,  taking  /-  =  .5  and  ja  =.04. 

Take  A'  =  radius  of  division  circle  =  19^  ft.     Then 

h\  =  19^  cos  12°  +  20  cos  40°  —  34.3947662  ft., 
and  hi  =  20  cos  20°  —     20  cos  40°  =  3.472964  ft. 

Therefore  the  H.P.  due  to  weight 

62^  .  400/  i  f  \ 

33000    V34'  3947662  +  -  x  3-472964J 

=  27.37215, 
and  the  H.P.  due  to  impact 

62$      400 

=  —  .  -  9(16  cos  10   —  9) 
32     33000  ^ 


=  1.43968. 

Again,  the  weight  of  the  water  on  the  wheel 

i     20  \ 

-  .  - 

2  i  so) 

=  2231.04  Ibs.,  approx., 


400.  62^/        128 

=  —  -     207T  —  —   +   207T  .  -  .  - 

60.9  v     i  so 


and  the  total  weight  on  the  axle  =  22231.04  Ibs. 

Thus  the  energy  absorbed  by  frictional  resistance  in  H.P. 

22231.04  i 

=  -        —  x  .04  x  9—  =  .18189, 
550  V 

and  hence 

the  net  useful  work  in  H.P.  =  27.37215  +  1.43968  —  .18189 

=  28.62994. 

The  total  available  H.P. 


.  400  /  i62 


33000 


•-    +   20  COS   12°    +   20 =   33.0022, 


•)- 


28  6^994 
and  therefore  the  efficiency  =  — —222.  —  .8675. 


47  2 


PITCH-BACK   WHEEL. 


17.  A  pitch-back  or  high  breast  wheel  is  to  be  preferred 
to  an  overshot  wheel  when  the  surface-levels  of  the  head-  and 
tail-water  are  liable  to  very  considerable  variation. 

In  the  pitch-back  wheel  the  water  is  admitted  by  aft 
adjustable  sluice  into  the  buckets  on  the  same  side  as  the 
supply-channel,  Figs.  274  and  275.  Thus  the  wheel  revolves 


FIG.  274. 


FIG.  275. 


FIG.  276. 


in  the  direction  in  which  the  water  leaves,  and  the  drowning 
of  the  wheel  is  prevented.  Further,  the  buckets  may  be  now 
ventilated,  Fig.  277,  and  may  therefore  be  placed  closer 
together  than  in  the  un ventilated  overshot  wheel. 

The  efficiency  of  the  pitch-back  is  at  least  equal  to  that  of 
the  overshot. 


EXAMPLES.  473 


EXAMPLES. 

1.  An  undershot  wheel  works  in  a  rectangular  channel  4  ft.  wide,  in 
•which  the  water  on  the  up-stream  side  is  2  ft.  deep  and  flows  with  a 
velocity  of  12  ft.  per  second  ;   the  water  on  the  down-stream  side  is  3  ft. 
deep.     Find  the  useful  work  done  and  the  efficiency. 

Ans.   1000  ft.-lbs.;  /T. 

2.  Determine  the  maximum  mechanical  effect  of  an  undershot  wheel 
of  12  ft.  diameter  making  10  revolutions  per  minute,  the  fall  being  3  ft. 
and  the  quantity  of  water  passed  per  second  15  cu.  ft. 

Ans.  1423  ft.-lbs. 

3.  Ascertain   the   general   proportions    of   a    Poncelet    wheel,  being 
given  :  height  of  fall  =  4^  ft.;  delivery  of  water  =  40  cu.  ft.  per  second  ; 
radius  of  exterior  circumference  =  9  ft.;  y  —  20°. 

Ans.  a.  =  143°  57';  ip  =  128°.  i  ;  d  •=.  2.  ft.;  ;-'  =  2.47  ft.; 
A.  =  1 5°.2  ;  /  =  5  ins.;  N  =  57  ;  tj  —  .69. 

4.  Design  a  Poncelet  wheel  for  a  fall  of  4.5  ft.  and  24  cu.  ft.  of  water 
per  second,  using  the  formulae  on  pages  428-432,  taking  y  =  20°,  and 
also  A  i=  20°  as  a  first  approximation. 

Ans.  <*  =  143°  57';  depth  of  crown  =  1.8  ft.;  depth  of  stream 
=  .372  ft  ;  £  =  4.14  ft.;  radius  of  bucket  —  2.26  ft.;  ^  =  128°  6'; 
A  =  17°  i';  number  of  buckets  =  48  ;  mechanical  effect  =  8.5  H.P.; 
efficiency  =  .69. 

5.  An  undershot  water-wheel  with  straight  floats  weighing  1 5,000  Ibs. 
works  in  a  straight  rectangular  channel  of  the  same  width  as  the  wheel, 
viz.,  4  ft.;  the  stream  delivers  28  cu.   ft.  of  water  per  second,  and  the 
efficiency  is  ^.      Find   the  relation   between   the  up-stream  and  down- 
stream  velocities.      If  the  velocity  of  the  inflowing  water  is  20  ft.  per 
second,  find  the  velocity  on  the  down-stream  side  and  determine  the 
mechanical  effect  of  the  wheel,  its  diameter  being  20  ft.,  the  diameter  of 
the  gudgeons  being  4  ins.,  and  the  coefficient  of  friction  .008. 

Ans.  3634.06  ft.-lbs. 

6.  Determine  the  effect  of  a  low  breast  or  undershot  wheel  15  ft.  in 
•diameter  and  making  8  revolutions  per  minute,  the  fall  is  4  ft.  and  the 
delivery  20  cu.  ft.  per  second  ;  the  velocity  of  the  stream  before  coming, 
on  the  wheel  is  double  that  of  the  wheel.  Ans.  3148  ft.-lbs. 

7.  20  cu.  ft.  of  water  per  second  enter  an  undershot  wheel  of  30  ft. 
diameter,    making    8    revolutions    per   minute,  through    an    underflow 
sluice.     The  velocity  of  the  entering  water  is  twice  that  of  the  wheel's 
periphery.     Find  (d)  the  head  of  water  behind  the  sluice;  (b}  the  fall;  (c) 
the  theoretical  mechanical  effect;  (d)  the  actual  mechanical  effect,  disre- 
garding axle-friction. 


474  EXAMPLES. 

Ans.  (a)  2.716  ft.;  (&)  1.283  ft-;  fc)  5-72  H.P.;  (</)  2.69  H.P. 

8.  20  cu.  ft.  of  water  per  second  enter  an  undershot  wheel  of  20  ft. 
diameter  in  a  straight  race,  the  fall  being  3  ft.     The  depth  of  the  enter- 
ing stream  is  £  ft.     The  width  of  the  wheel  is  4!  ft.,  and  the  clearance  is 
f   inch.     The  number  of  the  floats,  of  which  four  are  immersed,  is  48, 
and  each  is  i  ft.  long.     The  weight  of  the  wheel  is  7200  Ibs.,  the  radius 
of  the  axle  is  if  ins.,  and  the  coefficient  of  friction   is  .1.     Find  (a)  the 
best  speed  for  the  wheel ;  (b]  the  corresponding  mechanical  effect ;  (c]  the 
efficiency. 

•    Ans.  (a)  6  ft.  per  second  ;  (b)  2.32  H.P.,  assuming  the  speed  of 
wheel  reduced  to  5.74  ft.  per  second  by  axle-friction  ;  (c}  .34. 

9.  72  cu.  ft.  of  water  are  delivered  to  an  undershot  wheel  with  straight 
floats,  through  a  channel  of  rectangular  section  and  5  ft.  wide.     The 
velocity  (z/i)  of  the  inflowing  water  is  24  feet  per  second.    If  the  efficiency 
of  the  wheel  is  .25,  show  that  the  peripheral  speed  (u)  of  the  wheel  must 
be  6  ft.  per  second.     Also  determine  the  mechanical  effect  of  the  wheel, 

Ans.  10.125  ft.-lbs.  per  second. 

10.  The  water  in  a  rectangular  channel,  of  constant  width,  is  il  ft. 
deep,  and  impinges  upon  the  flat  buckets  of  an   undershot  wheel  with  a 
velocity  of  12  ft.  per  sec.     Show  that  the  efficiency  is  greatest  and  equal; 
to  .181  for  a  peripheral  speed  of  8.842  ft.  per  sec. 

11.  Water  enters  the  buckets  of  a  low  breast-wheel  with  a  velocity  of 
10  ft.  per  sec.  and  in  a  direction  making  an  angle  of  27°  44'  witli  the  tan- 
gent at  the  point  of  entrance,  which  is  4  ft.  measured  horizontally  and 
2  ft.  measured  vertically  from  the  sluice  where  the  stream-lines  are  hori- 
zontal.    Each  cubic  foot  of  water  does  563^  ft.-lbs.  of  useful  work  per 
sec.  when  the  wheel  makes  4f  revols.  per  min.     Find  the  fall  on  the 
wheel,  the  total  available  fall,  and  the  diam.  of  the  wheel. 

Ans.  8.437  ft.;   10  ft.;  24  ft. 

12.  A  race  is  straight  and  close-fitting  so  that  the  loss  of  effect  due 
to  escape  of  water  may  be  disregarded.     A  single  undershot  wheel  with 
plane  floats  is  replaced  by  four  similar  tandem  wheels.     If  the  delivery 
of  each  of  the  four  wheels  is  the  same,  and  if  it  is  assumed  that  the 
water  reaches  each  wheel  with  the  same  velocity  with  which  it  leaves  the 
preceding  wheel,  find  the  total  maximum  delivery  due  to  impact. 

Ans.  i£  times  the  delivery  of  the  single  wheel. 

13.  Discuss  the  preceding  example,   assuming  that  the    delivery  of 
each  wheel  is  not  the  same,  but  that  the  total  delivery  is  a  maximum. 

Ans.   1.6  times  the  delivery  of  the  single  wheel. 

14.  If  «  wheels  of  the  same  type  are  substituted  for  the  single  wheel 
in  example  12,  and  if  the  assumptions  are  the  same  as  those  in  example 
13,  show  that  the  total  delivery  of  the  n  wheels  is  to  the  delivery  of  the 
single  wheel  in  the  ratio  of  2«  to  2«  -f  i.  and  that,  theoretically,  if  the 
number  is  made  very  large,  they  will  approximately  give  the  entire  wortc 
of  the  fall. 


EXAMPLES.  475 

15.  In  a  low  breast-wheel  of  20  ft.  diameter,  the  water  enters  the 
bucket  with  a  velocity  of  16  ft.  per  second  in  a  direction  making  angles 
of  45°  with  the  horizontal  and  15°  with  the  wheel's   periphery.     The 
wheel  makes  7  revolutions  per  minute  and  receives  5  cu.  ft.  per  second 
of  water.     Find  the  mechanical  effect  of  the  wheel  and  the  position  of 
the  sluice,  which  is  placed  where  the  stream-lines  are  horizontal. 

Ans.  2075  ft.-lbs.;  AD  —  2  ft.,  BD  =-.  .25  ft. 

1 6.  The  water  in  a  head-race  stands  4.66  ft.  above  the  sole  and  leaves 
the  race  under  agate  which  is  raised  6  ins.  above  the  sole,  the  coefficient 
of  velocity  (vi)  being  .95.     The  water  enters  a  breast-wheel  in  a  direction 
making  an  angle  of  30°  with  the  tangent  to  the  wheel's  periphery  at  the 
point  of  entrance.     The  speed  («)  of  the  periphery  is  10  ft.  per  second, 
the  breadth  of  the  wheel  is  5  ft.,  the  depth  of  the  water  in  the  flume  is. 
8  ins.,  and  the  length  of  the  flume  is  8. 2  ft.    Find  the  loss  of  head  (a)  due 
to  the  destruction  of  the  relative  velocity  ( V)  at  entrance ;  (b)  due  to  the 
velocity  of  flow  in  the  tail-race  ;  (c)  in  the  circular  flume.     (/  =  .018.) 

Ans.  (a)  1. 1 1  ft.;  (b)  1.57  ft.;  (c)  .44  ft. 

17.  In  the  preceding  example,  find  how  the  losses  of  head  would  be 
modified  if  the  flume  were  lowered  1.03  ft.,  and  if  the  point  of  entrance 
were  raised  so  as  to  make  u  =  Vi  cos  30°. 

Ans.  (a)  .939  ft.;  (b)  2.816  ft.;  (c)  146  ft. 

1 8.  20  cu.  ft.  of  water  per  second  eriter  a  breast-wheel  of  32  ft.  diam- 
eter and  having  a  peripheral  velocity  of  8  ft.  per  second,  at  an  angle  of 
25^°  with  the  circumference.    The  depth  of  the  crown  is  i  J  ft.;  the  buck- 
ets  are  half-filled,  and   the  fall  is   9  ft.     The  velocity  of  the  entering 
water  is  12  ft.  per  second.     The  centre  of  the  sluice-opening  is  .54  ft. 
above  the  point  of  entrance,  and  the  width  of  the  sluice  is  3!  ft.     The 
wheel  has  48  buckets.     The  distance  between  the  wheel  and  breast  is  \ 
inch.     The  bucket  passes  through  .9  ft.  while  receiving  water,  and  the 
depth  of  the  water-surface  in  the  bucket  below  the  point  of  entrance  is 
1.25  ft.     Find  (a)  the  angular  distance  of  the  point  of  entrance  from  the 
horizontal ;  (b)  the  fall  in  the  breast ;  (c)  the  head  of  water  over  the  sluice; 

(d)  the  velocity  of  the  water  in  the  bucket  the  moment  entrance  ceases; 

(e)  the  total  mechanical  effect,  disregarding  axle-friction. 

Ans.  (a)  53°  53' ;  (^6.525  ft.;  (c)  1.935  ft.;  (^)  i4-9ft.;(*)  15.50!!.?. 

19.  In  the  preceding  question,  if  the  energy  absorbed  by  axle-friction, 
etc.,  is  743  ft.-lbs.,  find  the  efficiency  of  the  wheel.  Ans.  f . 

20.  15  cu.  ft.  of  water  per  second  with  a  fall  of  8|  ft.  are  brought  on 
a  breast-wheel  revolving  with  a  "linear  velocity  of  5  ft.;  depth  of  shroud- 
ing =  12  in.;  the  buckets  are  half-filled,  and  z/i  —  in ;  also  rl  =  12  ft. 
Find  the  theoretical  mechanical  effect,  y  being  30°.     Ans.  7040  ft.-lbs; 

21.  A  wheel  is  to  be  constructed  for  a  3o-ft.  fall  having  an  8-ft.  veloc- 
ity at  circumference  and  taking  on  the  water  at  12°  from  the  summit 
with  a  velocity  of  16  ft.     Determine  the  radius  of  the  wheel  and  the 
number  of  revolutions,  7/1  being  2n.  Ans.  12.9  ft.;  5.9. 


476  EXAMPLES. 

22.  If  for  the  wheel  in  example  21  the  number  of  revolutions  is  5, 
and  vi  =2u,  the  water  being  again  taken  on  at  12°,  find  the  radius  and  u. 

Ans.  13.56  ft.;  7.1  ft.  per  second. 

23.  A  breast-wheel  passes  12  cu.  ft.  of  water  per  second,  and  for  the 
speed  u  =  %Vi  =  4  ft.  per  second,  the  loss  of  mechanical  effect,  due  to  the 
relative  velocity  V  being  destroyed,  is  a  minimum.     Find  this  effect, 
y  being  30°.  Ans.  73.2  ft.-lbs. 

24.  In  a  breast-wheel  Q  =  10  cu.  ft.  per  second  ;  H  =  10  ft.;  ?/t  = 
J-#  ;  u  —  4^  ft.  per  second  ;  y  =  30° ;  diameter  of  gudgeon  =  6  ins.;  diam- 
eter of  wheel  =  30  ft.;  n  —  .08  ;  weight  of  wheel  and  water  =  20,000 
Ibs.     Find  the  mechanical  effect  of  the  wheel.     (Neglect  loss  of  effect 
due  to  escape  of  water  from  buckets  and  to  frictional  resistance  along 
the  curb.)  Ans.  5776  ft.-lbs. 

25.  The  quantity  of  water  laid  on  a  breast-wheel  by  an  overfall  sluice 
=  6  cu.  ft.  per  second,  the  total  fall  being  8  ft.,  and  the  velocity  of  the 
periphery  5  ft.  per  second  ;  also  5^1  =  Su,  and  if  d  be  the  depth  of  the 
shrouding  zbdit  =  $Q  (in  the  present  case  d  =.  12  ins.).    Find  the  effective 
fall,  the  height  of  the  lip  of  the  guide,  the  angle  of  inclination  at  the  end 
of  the  guide-curve,  the  breadth  of  the  lip  of  the  guide-curve,  and  the* 
radius  of  the  wheel  that  the  water  may  enter  tangentially.    If  the  radius 
is  limited  to  12  ft.  6  ins.,  find  the^  deviation  of  the  direction  of  motion  of 
the  water  from  that  of  the  wheel  at  the  point  of  entrance,  c  being  .6. 

Ans.  6.9  ft.;  .325  ft.;  34°  46';  2|  ft.;  38.6  ft.;  28°  36'. 

26.  10  cu.  ft.  of  water  per  second  are  delivered  to  a  breast-wheel. 
The  total  fall  is  10  ft.     The  peripheral  velocity  of  the  wheel  is  6  ft.  per 
second.     If  7-1  =  iu  and  y  —  30°,  find  the  theoretical  useful  effect  and 
the  theoretical  efficiency. 

Ans.  5358.4375  ft.-lbs.;  .85735. 

27.  24  cu.  ft.  of  water  enter  the  buckets  of  a  36-ft.  breast-wheel,  the 
total  fall  being  uj  ft.     At  the  point  of  entrance  the  direction  of  the 
water  makes  an  angle  of  30°  with  the  periphery  and  also  27/1  —  $u.    Find 
the   mechanical   effect  of  the  wheel  and  the  position  of  the  lip  of  the 
sluice  through  which  the  water  passes  to  the  wheel. 

Also,  if  the  depth  of  the  shrouding  is  i  ft.  and  the  buckets  are  only 
half-filled,  find  the  width  of  the  wheel. 

The  axle-bearings  are  6  ins.  in  diameter.  Taking  the  coefficient  of 
friction  to  be  .008,  how  much  power  is  absorbed  by  frictional  resistance, 
assuming  the  weight  of  the  wheel  and  contents  to  be  30,000  Ibs.? 

Ans.  26.833  H.P.;  x  —  .1624  ft.;  y  —  .5625  ft.;  10  ft.;  4.19  H.P. 

28.  In  an  overshot  wheel  rl  =  15   ft.,  d  —  10  in.,   ft  =  f^.     If  the 
division  circle  is  at  one  half  of  the  depth  of  the  crown,  find  the  angle 
(Xi)  between  the  bucket-lip  and  the  wheel's  periphery.    (Take  Ar  =  5^,.) 

Ans.  y\  =  1 8°  2'. 

29.  An  overshot  wheel,  in  which  r,  =  18  ft.,  makes  4  revolutions  per 
minute,  and  the  velocity  of  the  water  on  entering  the  buckets  is  twice 


EXAMPLES.  477 

that  of  the  wheel's  periphery.    If  y\  =  20°,  find  y,  and  also  find  the  rela- 
tive velocity  ( V)  of  the  entering  water. 

Am.  y  =  10°  9'  ;    V  —  7.78  ft.  per  second. 

30.  If  one  fourth  of  the  theoretic  capacity  of  a  bucket  is  filled  by  the 
water,  find  the  greatest  number  of  buckets  theoretically  possible,  the 
depth  of  the  crown  being  i  ft.,  the  radius  (r,)  to  the  outer  periphery  12 
ft.,  the  angle  yri  20°,  and  the  velocity  of  the  entering  water  twice  that  of 
the  wheel's  periphery. 

Ans.  103.1.     Making  allowance  for  exit  of  air,  the  number  of 
buckets  might  be  about  two  thirds  of  this  amount,  or,  say,  69. 

31.  A  wheel  of  30  ft.  diameter  with  72  buckets  makes  7  revolutions 
per  minute,  Q  being  5  cu.  ft.  per  second.    The  division  circle  is  half  way 
between  the  outer  and  inner  peripheries.     If  d  —  i  ft.  and  v\  =  2u,  find 
the  effect  due  to  impact.  Ans.  514  ft.-lbs. 

32.  A  3o-ft.  wheel  weighs  24,000  Ibs.  and  makes  6  revolutions  per 
minute;  its  gudgeons  are  6  ins.  in  diameter  and  the  coefficient  of  friction 
is  .08.     The  water  enters  the  wheel  with  a  velocity  of  15  ft.  per  second, 
and  in  a  direction  making  an  angle  of  10°  with  the  direction  of  motion 
of  the  wheel  at  the  point  of  entrance.     The  deviation  from  the  summit 
of  the  point  of  entrance  is  12°,  of  the  point  where  spilling  begins  is  150°, 
of  the  point  where  all  is  spilt  is  160°,  and   5  cu.  ft.  of  water  enter  the 
wheel  per  second,  of  which  the  partially  filled  buckets  contain  one  half. 
Determine  the  total  mechanical  effect.  Ans.  9114  ft.-lbs. 

33.  The  velocity  of  the  outer  periphery  is  9!  ft.;  the  angle  between 
the  directions  of  motion  of  stream  and  wheel  is  15°.     Find  the  impulsive 
effect  of  the  water,  7/1  being  15  ft.  per  second. 

Ans.  91  ft.-lbs.  per  cu.  ft.  of  water. 

34.  An   overshot  wheel  40  ft.  in  diameter  makes  4  revolutions  per 
minute  and  passes  300  cu.  ft.  of  water  per  minute.     If  the  gudgeons  are 
6  ins.  in  diameter  and  the  wheel  weighs  30,000  Ibs.,  by  how  much  will  the 
mechanical  effect  be  diminished?     (f  =  .008.) 

Ans.  25  ft.-ibs.  per  second. 

35.  The  diameter  of  an  overshot  wheel  =  30  ft.;  v\  —  15  ft.;  u  =  9$ 
ft.;  deviation  of   impinging  water  from  direction    of  motion  of  wheel 
(y)  =  81° ;  deviation  of  point  of  entrance  from  summit  =  12° ;  deviation 
of  point  where  spilling  begins  from  the  centre  =  58^°;  deviation  of  point 
where  spilling  ends  =  70*° ;  Q=$  cu.  ft.     Find  total  effect  of  impact 
and  weight.  Ans.  16.9  H.P. 

36.  An   overshot  wheel  with  a  radius  of   15  ft.  and  a  12- in.  crown 
takes  10  cu.  ft.  of  water  per  second  and  makes  5  revolutions  per  minute. 
If  m  =  J,  find  the  width  of  the  wheel  and  the  number  of  the  buckets. 

Ans.  5^  ft.;  75  or  90. 

37.  An  overshot  wheel  of  32  ft.  diameter  makes  5  revolutions  per 
minute.     Find  the  angle  between  the  water-surface  in  a  bucket  and  the 
horizontal  when  the  lip  is  140°  from  the  summit.  Ans.  4°  33'. 


478  EXAMPLES. 

38.  An  overshot  wheel  of  10  ft.  diameter  makes  20  revolutions  per 
minute.      Find  the  angle  between  the  water-surface  and  the  horizontal 
when  the  lip  is  (i)  90°  from  the  summit,  (2)  45°  26'  from  the  summit. 

Ans.  (i)  34°  27';  (2)43°  18'. 

39.  The  water  enters  an  overshot  wheel  at  12°  from  the  summit  with 
a  velocity  of  16  ft.  per  second  and  the  linear  velocity  of  the  wheel's  pe- 
riphery is  8  ft.  per  second.     The  fall  is  30  ft.     Find  the  diameter  of  the 
wheel  and  the  number  of  revolutions,  per  minute.      Ans.  25.4  ft.;  5.95. 

40.  In  a  32-ft.  wheel  with  a  12-in.  crown  and  a  peripheral  velocity  of 
8   ft.   per   second,  the  point   where   spilling   commences    is  defined  by 
6  =  0.     Find  the  arc  over  which  spilling  takes  place,  the  angle  between 
the  bucket-arm  and  the  circumference  being  30°.     Also  find  the  bucket- 
angle.    If  ii  cu.  ft.  of  water  enter  the  wheel  at  15°  from  the  summit  with 
a  velocity  of  18  ft.   per  second,  find  the  mechanical  effect  due  to  im- 
pulse and  to  weight,  k  being  4. 

Ans.  arc  =  15.2  ft.;  ft  =  56°  35';  2.87  H.P.,  28  H.P. 

41.  An  overshot  wheel  of  32  ft.  diameter  revolves  with  an  angular 
velocity  GO  •  show  that  the  angle  between  the  horizontal  and  the  water- 
surface  in  a  bucket  at  90°  from  the  summit  is  tan-1  — . 

42.  A  water-wheel  has  an  internal  diameter  of  4  ft.  and  an  external 
diameter  of  8  ft.;  the  direction  of  the  entering  water  makes  an  angle  of 
15°  with  the  tangent  to  the  circumference.      Find  the  angle  subtended 

^t  the  centre  of  the  wheel  by  the  bucket,  which  is  in  the  form  of  a  circu- 
lar arc,  and  also  find  the  radius  of  the  bucket.  Ans.  28°  54' ;  1.2274  ft. 

43.  An  overshot  wheel  5  ft.  wide,  30  ft.  in  diameter,  having  a  12-in. 
crown  and  72  buckets,  receives  10  cu.  ft.  of  water  per  second  and  makes 
5  revolutions  per  minute.     Determine  the  deviation  from  the  horizontal 
-at  which  the  water  begins  to  spill,  and  also  the  corresponding  depres- 
sion of  the  water-surface.  Ans.  31°  41'  ;  5°  51'. 

44.  An  overshot  wheel  makes  —  revolutions  per  minute  ;  its  mean 

diameter  is  32  ft.;  the  water  enters  the  buckets  with  a  velocity  of  8  ft. 
per  second  at  a  point  12°  30'  from  the  summit  of  the  wheel.  At  the 
point  of  entrance  the  path  of  the  inflowing  water  makes  an  angle  of  30° 
with  the  horizontal.  Show  that  the  path  is  horizontal  vertically  above 
the  centre.  The  sluice-board  is  placed  at  a  point  whose  horizontal  dis- 
tance from  the  centre  is  one  half  that  of  the  point  of  entrance.  Find  its 
position  relatively  to  the  centre  and  its  inclination  to  the  horizon. 
Also  find  V.  Ans.  16°  6' ;  6.24  ft.  per  second. 

45.  The   water  enters  the  buckets  of  the  wheel  in   the    preceding 
example  without  shock.     Find  the  elbow-angle.     Also,  if  the  buckets 
begin  to  spill  at  150°  from  the  summit,  find  where  the  bucket  is  empty 
and  the  number  of  buckets.     (Depth  of  crown  =  12.  ins.;   thickness  of 
bucket  =  \\  ins.)  Ans.  125°  30';  156°  10';  80. 


EXAMPLES.  479 

46.  Given  7/1  =  15  ft.  per  second,  and  d  =  "20^°.     Find  the  position 
of  the  centre  of  the  sluice,  which  is  4  ins.  above  the  point  of  entrance. 

Ans.  .0877  ft.  vertically  below  and  1.0143  ft-  horizontally  from 
the  summit.  The  axis  of  the  sluice  is  inclined  at  9°  33'  to  the 
horizontal.  (Assume  y  =o°.) 

47.  In  an 'overshot  water-wheel  v\  —  15  ft.;  u  —  10  ft.;  elbow-angle 
=  7oi°;    division-angle  =  4^°;  deviation  from   summit  of  point    of    en- 
trance =  12°.     Find  the  deviation  of  the  layer  from  that  of  the  arm,  so 
that  the  water  might  enter  unimpeded  ;  also  find  the  inclination  of  the 
layer  to  the  horizon,  and  the  value  of  V.     If  the  centre  of  the  sluice- 
aperture  is  to  be  4  ins.  above  point  of  entrance,  find  its  vertical  and 
horizontal  distance  from  the  vertex  of  the  stream's  parabolic  path  which 
is  vertically  above  the  centre  of  the  wheel,  and  also  find  inclination  of 
sluice-board  to  horizon. 

Ans.   5!° ;  20^°  ;  5.3  ft.  per  second  ;  .0878  ft.;   1.04  ft.;  9°  34'. 

48.  A  wheel  makes  20  revolutions  per  minute  ;  radius   -  5  ft.,  angle 
of  discharge  =  o°.     Frnd  deviation  of  water-surface  from  horizon.    Also 
find  deviation  at  44°  35'  above  centre.  Ans.  4°  27' ;  43°  16'. 

49.  In  an  overshot  wheel  Q  =  18  cu.  ft.  ;  r\  =  6  ft.  ;  d  —  \  ft.;  b  =  4  ft.; 
JV  =  24  ;  11  —  17.     At  the  moment  spilling  commences  the  area  cbfd  = 
1.025  sq.  ft.;  between  this  point  arid  the  point  where  the  spilling  is  com- 
pleted three  buckets  are  interposed,  the  sectional   areas  of  the  water 
being  .501,  .409,  and  .195  sq.   ft.,  respectively.      Find  (a)  the  sectional 
area  of  bucket ;  (b)  the  point  where  the  spilling  commences ;  (c)  the  point 
where  the  spilling  is  completed  ;  (d)  the  height  of  the  arc  of  discharge ; 
(e)  the  mechanical  effect  due  to  the  fall  of  the  water  through  the  arc  of 
discharge,  y  being  10°  46'. 

Ans.  (a)  .662  sq.  ft.  ;  (b)  Q  =  7°  13',  0  =  28°  46'  ;  (c)  0  =  73°  23', 

<t>=  5°  51';  <X)  4-49  ft-;  0)  4.93  H.P. 

50.  In  the  preceding  example,  if  the  water  enters  with  a  velocity  of 
20  ft.  per  second  at  20°  below  the  summit,  and  if  the  direction  of  the 
inflowing  stream  makes  an  angle  of  25°  with  the  wheel's  periphery  at 
the   point  of  entrance,  find  the  mechanical  effect  (a)   due  to   impulse  ; 
(b)  due  to  the  fall  to  the  point  where  spilling  commences. 

Ans.  (a)  5.08  H.P.  ;  (b)  12.114  H.P. 

51.  300  cu.  ft.  of  water  per  minute  enter  the  buckets  of  a  4o-ft.  over- 
shot wheel  with  a  12-in.  crown  and  making  four  revolutions  per  minute. 
The  wheel  has  136  buckets.     At  the  moment  when  spilling  commences 
the  area  bcdf  —  126.5  sq.  in.     The  spilling  is  completed  when  the  angle 
between  the  horizontal  and  the  radius  to  the  lip  of  the  bucket  =  62°  30'. 
Between  these  two  positions  three  buckets  are  interposed,  the  sectional 
areas  of  the  water  in  the  buckets  being  24.5,  14.48,  and  6.6  sq.  ins.,  respec- 
tively.    The  vertical  distance  between    the   water-surface    in    the  first 
bucket  and  the  centre  is  18  ft.     Find  (a)  the  width  of  the  wheel ;  (b)  the 
cross-section  of  a  bucket ;  (c)  the  angle  between  the  horizontal  and  the 


480  EXAMPLES. 

radius  to  the  lip  of  the  bucket  when  spilling  commences  ;  (d)  the  height 
of  the  discharging  arc  ;  (<?)  the  mechanical  effect  due  to  weight. 

Ans.  (a)  2.4  ft.  ;  (£)  33.28  sq.   ft.  ;  (V)  0  =  52°  19'  ;  (//)   1.9  ft.  ; 
(*)  19-73  H.P. 

52.  As   the  bucket-arm  cd  moves  downward    from    the    horizontal 
position,  show  that  while  the  wheel  moves  through  an  angle  6  the  last 
particle  of  water  at  c  will  move  through  a  distance  approximately  equal 

to  r^r  *  u  '-  (Q  —  sin  6),  r  being  the  distance  (assumed  constant)  of  the 

particle  of  water  from  the  axis,  and  u  being  the  linear  velocity  of  the 
wheel  at  the  radius. 

53.  If  the  last  particle  of  water  leaves  the  buckets  just  as  the  lip  d 
reaches  the  lowest  point  of  the  wheel,  and  if  the  arm  is  I  ft.  in  length, 
find  the  angle  between  the  lip  and  the  wheel's  periphery  (i)  for  a  wheel 
of  20  ft.  diameter,  the  peripheral  velocity  being  5  ft.    per  second  ;  (2)  for 
a  wheel  of  40  ft.  diameter,  the  peripheral  velocity  being  loft,  per  second  ; 
(3)  for  a  wheel  of  10  ft.  diameter,  the  peripheral  velocity  being  8  ft.  per 
second.  Ans.  (i)  20° ;  (2)  19.5°;  (3)  40°. 

54.  In  an  overshot  wheel  of  30  ft.  diameter,  5  cu.  ft.  of  water  per 
second  enter  the  buckets  with  a  velocity  of  16  ft.  per  second  and  the 
wheel's  velocity  at  the  division  circle  is  7  ft.  per  second.     The  point  of 
entrance  is  18°  from  the  summit,  and  the  angle  between  the  directions 
of  the  inflowing  water  and  the  wheel's  periphery  at  the  point  of  entrance 
is  12°.     The  water  begins  to  spill  at    148^°  from  the  summit  and  the 
spilling  is  complete  at  i6o|°  from  the  summit.     Find  the  total  mechan- 
ical effect  due  to  impulse  and  weight.     What  is  the  tangential  force  at 
the  outer  periphery?  Ans.  16.28  H. P.;  1194155. 

55.  In  a  32-ft.  wheel,  with  a  i-ft.  crown  and  a  peripheral  velocity  of 
8  ft.  per  second,  the  point  where  spilling  commences  is  defined  by  the 
relation  6  =  0.     Find  the  arc  over  which  spilling  takes  place,  the  angle 
between  the  arm  and  circumference  being  30°.    Also  find  the  "  bucket  " 
angle.     If  11  cu.  ft.  of  water  enter  the  wheel  at  15  ins.  from  the  summit, 
and  with  a  velocity  of  18  ft.  per  second,  show  how  to  find  the  mechanical 
effect  due  to  impulse  and  that  due  to  weight. 

Ans.    53°  23';  4°,  i6i2ff  ft.-lbs.  ;  14,406.56  ft.-lbs  per  second. 

56.  An  overshot  wheel  of  32  ft.  diameter  makes  -1//-  revolutions  per 
minute.    Find  the  inclination  to  the  horizontal  of  the  water-surface  in  a 
bucket  at  90°  from  the  summit.     If  the  wheel  has  90  buckets  and  the 
arms  make  an  angle  of  22^°  with  the  periphery,  find  the  depth  of  the 
crown.  Ans.  7°  8';   1 1  ins. 

57.  An  overshot  wheel  of  32  ft.  diameter  makes  yy5  revolutions  per 
minute.     Find  the  inclination  to  the  horizontal  of  the  water-surface  in  a 
bucket  at  90°  from  the  summit.  If  the  wheel  has  90  buckets  and  the  arms 
make  an  angle  of  22^°  with  the  periphery,  find  the  depth  of  the  crown. 

Ans.  25.68  ft.  ;  5.94. 


EXAMPLES.  481 

58.  An  overshot  wheel  of  36  ft.  diameter  and  with  96  buckets  has  a 
peripheral  velocity  of  7$  ft.  per  second.    The  water  enters  with  a  velocity 
of  15  ft.  per  second  and  acquires  in  the  wheel  a  velocity  of  16.49  ft-  Per 
second.       Find   the    distance   through   which    the    float    moves  during 
impact.  Ans.  2.15  ft. 

59.  The  sluice  for  a  lo-ft.  overshot  wheel  is  vertically  above  the  cen- 
tre and  inclined  at  45°  to  the  vertical.     The  water  enters  the  buckets  at 
a  point  2  ft.  vertically  below  the  sluice  and  10°  from  the  summit  of  the 
wheel.    Find  the  angle  between  the  directions  of  motion  of  the  entering 
water  and  of  the  wheel's  circumference.     Also  find  the  velocity  of  the 
water  as  it  enters  the  wheel.  Ans.  5°  30' ;  9.68  ft.  per  second. 

60.  In  an  overshot  wheel  v\  =  17  ft.;  u  =  1 1   ft.  per  second  ;  elbow- 
angle  =  70°;  division-angle  =  5°;    water  enters  the  first  bucket  at   12° 
from  summit  of  wheel.      Find  (a)  the  relative  velocity   V  so  that  water 
may  enter  unimpeded;  (b)  the  direction  of  the  entering  water;  (c)  the 
diameter  of  the  wheel,  which  makes  5   revolutions  per  minute;  (d)  the 
position  and  direction  of  the  sluice,  which  is  2  ft.  measured  horizontally 
from  the  point  of  entrance. 

Ans.  6.24  ft.  per  second  ;  y  =  7°  13' ;  42  ft.;  45  ft.;  5°  43'. 

61.  In  an  overshot  wheel  the  deviation  of  the  impinging  water  from 
the  direction  of  motion  of  the  wheel  is  10°  ;  the  velocity  (v\)  of  the  im- 
pinging stream  —  15  ft.  per  second  ;  of  the  circumference  of  the  wheel 
(«)  ='15  cos  10°.     What  amount  of  the  head  is  sacrificed  ? 

Ans.  i. 06  ft. 

62.  A    3o-ft.   water-wheel  with  72  buckets  and    a    12-in.   shrouding 
makes  5  revolutions  and  receives  240  cu.  ft.  of  water  per  minute.     Find 
the  width  and  sectional  area  of  a  bucket.    The  fall  is  30  ft.;  at  what  point 
does  the  water  enter  the  wheel,  the  inflowing  velocity  being  i|  times 
that  of  the  wheel's  periphery  ?     Also  find  the  deviation  of  the  water- 
surface    from    the  horizontal  at  the  point   at  which  discharging  com- 
mences, i.e.,  140°  from  the  summit. 

Ans.  2.03'  ;  .327  sq.  ft.;  32°  47' ;  4°  18'. 

63.  What  number  of  buckets  should  be  given  to  an  overshot  wheel  of 
40  ft.  diameter  and  12  ins.  width  in  wheel,  pitch-angle  =  4°,  thickness  of 
bucket-lip  =  i  in.,  water  area  =  24^  sq.  ins.? 

Ans.  167,  depth  of  crown  being  9  ins. 


CHAPTER    VII. 
TURBINES. 

i.  Reaction  and  Impulse  Turbines.  —All  turbines  belong- 
to  one  of  two.  classes,  viz.,  Reaction  Turbines  and  Impulse 
Turbines,  and  are  designed  to  utilize  more  or  less  of  the  avail- 
able energy  of  a  moving  mass  of  water. 

In  a  reaction  turbine  a  portion  of  the  available  energy  is 
converted  into  kinetic  energy  at  the  inlet  surface  of  the  wheeL 
The  water  enters  the  wheel  passages  formed  by  suitably  curved 
vanes,  and  acts  upon  these  vanes  by  pressure,  causing  the 
wheel  to  rotate.  The  proportions  of  the  turbine  are  sucli  that 
there  is  a  particular  pressure  (hence  the  term  pressure-turbine) 
at  the  inlet  surface  corresponding  to  the  best  normal  condition 
of  working.  Any  variation  from  this  pressure,  caused,  e.g., 
by  the  partial  closure  of  the  passages  through  which  the  water 
passes  to  the  wheel,  changes  the  working  conditions  and 
diminishes  the  efficiency.  In  order  to  avoid  such  a  variation 
of  pressure,  it  is  essential  that  there  should  be  a  continuity  of 
flow  in  every  part  of  the  turbine ;  the  wheel  passages  should  be 
kept  completely  filled  with  water,  and  therefore  must  receive 
the  water  simultaneously.  Such  turbines  are  said  to  have 
complete  admission.  The  admission  is  partial  when  the 
water  is  received  over  a  portion  of  the  in'et  surface  only. 

In  an  impulse  (Girard)  turbine,  Figs.  277  and  278,  the 
energy  of  the  water  is  wholly  converted  into  kinetic  energy  at 
the  inlet  surface.  Thus  the  water  enters  the  wheel  with  a 
velocity  due  to  the  total  available  head  and  therefore  without 

482 


REACTION  AND  IMPULSE   TURBINES. 


48$ 


pressure,  is  received  upon  the  curved  vanes,  and  imparts  to  the 
wheel  the  whole  of  its  energy  by  means  of  the  impulse  due  to 
the  gradual  change  of  momentum.  Care  must  be  taken  to 
insure  that  the  water  may  be  freely  deviated  on  the  curved 
vanes,  and  hence  such  turbines  are  sometimes  called  turbines 
With  free  deviation.  For  this  reason  the  water-passages  should 


FIG.  277. 
Girard  Turbine  for  Low  Falls. 


FIG.  278. 
Girard  Turbine  for  High  Falls. 


never  be  completely  filled,  and  the  water  should  flow  through 
under  a  pressure  which  remains  constant.  In  order  to  insure 
an  unbroken  flow  through  the  wheel-passages  and  that  no 
eddies  are  formed  at  the  backs  of  the  vanes,  ventilating  holes 
are  arranged  in  the  wheel  sides,  Fig.  280.  Figs.  279  and  280 
also  show  the  relative  path  AB  and  the  absolute  path  CD 
traversed  by  the  water  in  an  inward-flow  and  a  downward-flow 
turbine. 

If  there  is  a  sufficient  head,  the  wheel  may  be  placed  clear 


4^4 


REACTION  AND  IMPULSE    TURBINES. 


above  the  tail-water,   when  the   stream   will   be   at  all   times 

under  atmospheric  pressure.      With 
low  falls  the  wheel  may  be  placed 
in  a  casing  supplied  with  air  from 
an  air-pump  by  which  the  surface 
of  the  water  may  be  kept  at  an  in- 
variable    level     below    the     outlet 
orifices,  which  is  essential  for  per- 
fectly   free    deviation.      While    the 
wheel-passages  of  a  reaction  turbine 
:should  be  kept  completely  filled  with  water,  no  such  restriction 
is   necessary  with  an   impulse   turbine.      The   supply  may  be 
partially  checked  and  the  water  may  be  received  by  one  or 


FIG. 


snore  vanes  without  affecting  the  efficiency.  Thus  the  dimen- 
sions of  an  impulse  turbine  may  vary  between  very  wide  limits, 
so  that  for  high  falls  with  a  small  supply  a  comparatively  large 
wheel  with  low  speed  may  be  employed.  The  speed  of  a 
-reaction  turbine  under  similar  conditions  would  be  disadvan- 
Uageously  great,  and  any  considerable  increase  of  the  diameter 
would  largely  increase  the  fluid  friction  and  would  also  render 
the  proper  proportioning  of  the  vane-angles  almost  impracti- 
cable. Impulse  turbines  may  have  complete  or  partial  admis- 
while  in  reaction  turbines  the  admission  should  be  always 


THE  HURDY-GURDY  AMD  PELT  ON   WHEEL. 


485 


complete,  as  in  Fig.  281,  which  shows  the  relative  path  AB 
and  absolute  path  CD  traversed  by  the  water.  When  there  is 
an  ample  supply  of  water  the  reaction  turbine  is  usually  to  be 
preferred,  but  on  very  high  falls  its  speed  becomes  incoii- 


WHEEL 


FIG.  281. 

veniently  great  and  it  is  then  better  to  adopt  a  turbine  of  the 
impulse  type.  The  diameter  of  the  wheel  can  then  be  increased 
and  the  speed  proportionately  diminished. 

The  Hurdy-gurdy  is  the  name  popularly  given  to  an 
impulse  wheel  which  was  introduced  into  the  mining  districts 
of  California  about  the  year  1865.  Around  the  periphery  of 
the  wheel  is  arranged  a  series  of  flat  iron  buckets,  about  4  to 
6  ins.  in  width,  which  are  struck  normally  by  a  jet  of  water 
often  not  more  than  three  eighths  of  an  inch  in  diameter. 
Theoretically  the  efficiency  of  such  an  arrangement  cannot 
exceed  50  per  cent,  while  in  practice  it  rarely  reaches  40  per 
cent.  The  best  speed  of  the  wheel,  in  accordance  with  both, 
theory  and  practice,  is  about  one  half  of  that  of  the  jet.  Although 
the  efficiency  is  so  low,  the  wheel  found  great  favor  for  many 
reasons.  Any  required  speed  could  be  obtained  by  a  suitable 
choice  of  diameter;  the  plane  of  the  wheel  could  be  placed  int. 
a*iy  convenient  position ;  the  wheel  could  be  cheaply  con- 
structed and  was  largely  free  from  liability  to  accident.  Hence 
it  was  of  the  utmost  importance  to  increase,  if  possible,  the 
efficiency  of  a  wheel  possessing  such  advantages. 


486 


ACTUAL   PATH  OF  FLUID  PARTICLE  IN    TURBINE. 


a  first  step  was  to  substitute  cups  for  the  flat  buckets,  the 
immediate  result  necessarily  being  a  very  large  increase  in  the 
efficiency.  This  was  increased  still  further  by  the  adoption  of 
double  buckets,  Fig.  282,  that  is,  curved  buckets  divided  in 
the  middle  so  that  the  water  is  equally  deflected  on  both  sides. 
Thus  developed,  the  wheel  is  widely  and  most  favorably 
known  as  the  Pelton  wheel,  Fig.  282.  Its  efficiency  is  at  least 
80  per  cent,  and  it  is  claimed  that  it  often  rises  above  90  per 
cent.  The  power  of  the  wheel  does  not  depend  upon  its 
diameter,  but  upon  the  available  quantity  and  head  of  water. 
The  water  passes  to  the  wheel  through  one  or  more  nozzles, 


FIG.  282. 

having  tips  bored  to  suit  any  required  delivery.  These  tips 
are  screwed  into  the  nozzles  and  can  be  easily  and  rapidly 
replaced  by  others  of  larger  or  smaller  size,  so  that  the  Pelton 
is  especially  well  adapted  for  a  varying  supply  of  water.  It  is 
claimed  that  in  this  manner  the  power  may  be  varied  from  a 
maximum  down  to  25  per  cent  of  the  same  without  appreciable 
loss  of  efficiency. 

2.  Actual  Path  of  a  Fluid  Particle  in  Passing  through 
a  Turbine. — Under  the  combined  effect  of  the  inlet  velocity 
i\  and  the  rotation  of  the  wheel,  a  fluid  particle,  entering  at  a, 
will  traverse  an  actual  path  of  cutting  the  outlet  surface  at  an 


ACTUAL   PATH  OF  FLUID  PARTICLE  IN   TURBINE. 


487 


angle  equal  to  d.      This  path  may  be  approximately  plotted 
in  the  following  manner: 

Let  of,  a'f  be  two  consecutive 
blades. 

Let  q  be  the  discharge  per  second 
through  the  passage  between  these 

blades. 

j 

Let  xx'  be  a  surface  concentric 
with  aa  and  of  radius  r. 

Let  t  be  the  time  in  seconds  in 
which  a  fluid  particle  flows  from  aa'  to 
jcx'. 

Let  A  be  the  area  axx 'a ' . 

Let  d  be  the  mean  depth  of  this 
area  between  the  crowns. 

Let  GO  be  the  angular  velocity  of 
the  wheel. 

Then    Qt  =  volume  of  water  between  aa'  and  xx'  =  Ad. 

But  in  the  same  time  /  the  point  x  will  have  moved  to  z 
where 


FIG.  283. 


xz  =  root  ==.  rood — . 


In  this  equation  the  values  of  GO  and  q  are  known,  so  that 
lay  describing  any  required  number  of  cylindrical  surfaces  and 
introducing  into  the  equation  the  corresponding  values  of  r,  d, 
and  A,  a  series  of  values  will  be  obtained  for  xs  defining  the 
points  zv ,  ^2 ,  &3  .  .  .  on  the  actual  path  al  of  the  fluid  particles. 

Zeuner  gives  a  somewhat  more  general  method  as  follows: 

Consider  a  fluid  particle  moving  along  the  axis  RM  of  the 
passage  between  two  consecutive  vanes  af  and  a'f . 

If  the  wheel  were  at  rest,  the  particle  in  /  seconds  would 
reach  a  certain  point  M,  but  the  rotation  of  the  wheel  carries 
it  to  M',  where  MM'  =  root,  r  being  the  radius  OM  (—  OM') 


488 


ACTUAL   PATH   OF  FLUID  PARTICLE  IN    TURBINE. 


and    GO  the    constant  angular    velocity  about    the   axis  at   O. 
Thus,  after  /  seconds,  M'  is  the  true  locus  of  the  fluid  particle. 


FIG.  284. 


Let  0  and  <p'  be  the  angular  deviations  of  OM  and  OM' 
from  OR. 

Let  ux  (—  M'P)  be  the  linear  velocity  of  M' . 

Let  Vx  (—  M'Q]  be  the  relative  velocity  of  the  fluid  particle 
at  M'. 

Let  vx  (=  M'N)  be  the  absolute  velocity  of  the  fluid  particle 
at  M'. 

Let  0  be  the  angle  between  V x  and  the  radius  OM  or  OM' + 

Let  &  be  the  angle  between  vx  and  OM'. 

Then 


vx  sin  0'  =  ux  —  Vx  sin  0 


and 


v    cos  tf  = 


cos 


ACTUAL   PATH  OF  FLUID  PARTICLE  IN    TURBINE.          489 
Hence 


cos 


n  =  tan  6  +  tan  6' 
6 


deb  dd>f 

But  ux  =  roo\  tan  6  —  r-^\  tan  0'  =  -r~- ,  since  M'N  is 


necessarily  tangential  to  the  actual  path  RM'  at  M' \  and 
Aj.Vj.cos  6  =  Q,  the  volume  of  flow  per  second,  Ax  being  the 
sectional  area  of  the  passage  at  right  angles  to  OM.  Substi- 
tuting these  values  in  the  last  equation, 

GO  d<t> 

^A^--^r^ 
and  therefore 


TJ  being  the  internal  radius  of  the  wheel. 

But  the  expression    /*  Adr  is   the  volume  of  the  passage 

t/r, 

between  aa'  and  M  and  may  be  determined  by  actual  measure- 
ment. 

Designating  this  volume  by  Ux ,  then, 


an  equation  giving  0  when  0r  is  known,  or  0'  when  0  is  known. 
Thus  the  actual  position  of  the  particle  can  be  determined  if 
its  relative  position  is  known,  or  its  relative  position  can  be 
found  when  its  actual  position  is  given. 


490  CLASSIFICATION  OF  TURBINES. 

Take  a  number  of  equidistant  points  M{ ,  M2 ,  M3  .  .  .  along 
the  axis  of  the  passage,  and  let  0X ,  02 ,  03  .  .  .  be  the  angular 
deviations  of  OMl ,  OM2 ,  <9J/3  .  .  .  from  OR. 

Also,  let  £/! ,  £/2 ,  £73  .  .  .  be  the  volumes  of  the  passage 
between  ##'  and  Ml ,  ##'  and  J^,  ##'  and  J^  .  .  .  Then  the 
angular  deviations  0/,  02',  03'  ...  of  the  radii  to  the  corre- 
sponding points  J/j',  J/2',  J/3'  ...  on  the  actual  path,  are 
given  by  the  equations 


and  the  actual  path  can  be  at  once  plotted. 

The  value  of   /  Axdr  can    easily   be    found    graphically. 

tA-j 

Thus,  plot  the  radii  OR,  OMlt  OM2  .  .  .  OM  as  abscissae,  and 
the  corresponding  sectional  areas  of  the  passage  at  -R,  Ml , 
.M2  ...  J/  as  ordinates.  Joining  the  upper  ends  of  these 
ordinates  by  a  suitable  curve,  the  area  between  this  curve,  the 
extreme  ordinates  and  the  line  of  abscissae  is  evidently  the 
volume  required.  This  area  may  be  determined  with  a  pla- 
nimeter. 

3.  Classification  of 'Turbines. — The  character  of  the  con- 
struction of  turbines  has  led  to  their  being  classified  as 
(i)  Radial-flow  turbines;  (2)  Axial-flow  turbines;  (3)  Mixed- 
flow  turbines. 

The  water  may  act  wholly  by  pressure  or  who  11}'  by  im- 
pulse, or  partly  by  pressure  and  partly  by  impulse,  or  by 
reaction.  In  pressure  wheels  the  water-passages  are  not  com- 


THE  FOURNEYRON   TURBINE.  49* 

pleteiy  filled  as  in  reaction  wheels.  In  impulse  wheels  the 
Water  spreads  out  in  all  directions,  while  in  pressure  and 
reaction  wheels  the  water  flows  off  on  one  side  only. 

In  Radial- flow  turbines  the  water  flows  through  the  wheel 
in  a  direction  at  right  angles  to  the  axis  of  rotation  and 
approximately  radial.  The  two  special  types  of  this  class  are 
the  Outward- flow  turbine,  invented  by  Fourneyron,  and  the 
Inward-flow  or  Vortex  turbine,  invented  by  James  Thomson. 
In  the  outward-flow  turbine.  Figs.  285  and  286,  the  water 
enters  a  cylindrical  chamber  and  is  led  by  means  of  fixed 
guide-blades  outwards  from  the  axis.  It  is  distributed  over 
the  inlet-surface,  passes  through  the  curved  passages  of  an 
annular  wheel  closely  surrounding  the  chamber,  and  is  finally 
discharged  at  the  outer  surface.  The  wheel  works  best  when 
it  is  placed  clear  above  the  tail-water.  A  serious  practical 
defect  is  the  difficulty  of  constructing  a  suitable  sluice  for  regu- 
lating the  supply  over  the  inlet-surface.  When  the  water  is 
insufficient  to  work  the  turbine  at  its  full  power,  the  exit 
openings  may  be  closed  to  any  required  extent  by  lowering 
a  cylindrical  sluice. 

A  well-designed  turbine  of  this  type  gives  an  efficiency  of 
70  per  cent,  and  the  maximum  efficiency  is  about  80  per  cent, 
but  the  efficiency  is  considerably  diminished  by  closing  the 
sluice.  Fourneyron  was  led  to  the  design  of  this  turbine  by 
observing  the  excessive  loss  of  energy  in  the  ordinary  Scotch 
turbine,  or  reaction  wheel,  and  introduced  guide-blades  in 
order  to  give  the  water  an  initial  forward  velocity  and  thus 
cause  a  diminution  of  the  velocity  of  the  water  leaving  the 
outlet-surface. 

Boyden's  turbine  is  a  modification  of  the  Fourneyron. 
The  water  is  conducted  to  the  guide-blades,  which  are  inclined 
so  as  to  receive  the  water  tangentially,  through  a  truncated 
cone;  and  the  water  thus  acquires  a  gradually  increasing 
velocity  together  with  a  spiral  motion.  The  wheel,  again,  is 
surrounded  by  a  diffusor  which  expands  outwardly  and  which 


492 


THE  BOY  DEN    TURBINE. 


should    be    completely    submerged.       The    water    then    flows 
through  the  wheel  with  an  increased  velocity  and  passes  away 

FIG.  285. 


™ 


FIG.  286. 

through  the  diffusor  with  a  velocity  which  gradually  diminishes. 
There  is  said  to  be  a  gain  of  3  per  cent  effected  by  this  arrange 


THE    YORTEX    TURBINE. 


493 


FIG.  287. 


ment,  while  Boyden  claimed  for  his  75 -H. P.  turbine  an  effi- 
ciency of  88  per  cent. 

In  the  Inward-flow  or  Vortex  turbine,  Figs.  287,  288,  and 

289,  the  wheel  is  en- 
closed in  an  annular 
space,  into  which  the 
water  flows  through  one 
or  more  pipes,  and  is 
usually  distributed  over 
the  inlet-surface  of  the 
wheel  by  means  of  four 
guide-blades.  The  water 
enters  the  wheel,  flows 
towards  the  space  around 
the  axis,  and  is  there 
discharged.  This  tur- 
bine possesses  the  great 
advantage  that  there  is 
ample  space  outside  the 

Thomson's  Vortex  Turbine. 


'///^///////"""W 


FIG.  288.  FIG.  289. 

wheel  for  a  perfect  system  of  regulating-sluices.      This  turbine 
has  attained  an  efficiency  of  77^  per  cent. 


494 


THE  AXIAL-FLOW   TURBINE. 


Axial-flow  turbines,  Fig.  290,  are  also  known  as  Parallel 
and  D(nvnward-flow  turbines  and  are  sometimes  called  by  the 
names  of  the  inventors,  Jonval  and  Fontaine.  In  these  the 
water  passes  downward  through  an  annular  casing  in  a  direc- 
tion parallel  to  the  axis  of  rotation,  and  is  distributed  by  means 
of  guide-blades  over  the  inlet-surface  of  an  adjacent  wheel.  It 
enters  the  wheel-passages  and  is  finally  discharged  vertically, 
or  nearly  so,  at  the  outlet-surface.  The  sluice-regulations  are 
worse  even  than  in  the  case  of  an  outward-flow  turbine,  but 
there  is  this  advantage,  that  the  turbine  may  be  placed  either 
below  the  tail-water,  or,  if  supplied  with  a  suction-pipe,  at  any 
point  not  exceeding  30  ft.  above  the  tail-water. 


FIG.  290. 

If  a  turbine  is  designed  so  that  the  pressure  at  the  clearance 
between  the  casing  and  the  wheel  is  nil,  and  with  curved 
passages  in  the  form  of  a  freely  deviated  stream,  it  becomes 
what  is  called  a  Limit  turbine.  In  its  normal  condition  of 
working  it  is  an  Impulse  turbine,  but  when  drowned  it  is  a 
Reaction  turbine,  with  a  small  pressure  at  the  clearance.  For 
moderate  falls  with  a  varying  supply  its  average  efficiency  is 
higher  than  that  of  a  pressure  turbine. 

The  Mixed-  or  Combined-flow  (Schiele)  turbine  is  a  com- 
bination of  the  radial  and  axial  types.  The  water  enters  in  a 


THE   AXIAL-FLOW   TURBINE.  495 

nearly  radial  direction  and  leaves  in  a  direction  approximately 
parallel  to  the  axis  of  rotation  This  type  of  turbine  admits 
of  a  good  mode  of  regulation  and  is  cheap  to  construct. 

The  Swain  turbine  is  a  combination  of  the  inward-  and 
axial-flow  types.  The  vane  inlet-lips  are  vertical  opposite  the 
guide-blades,  and  at  the  outlet  the  vanes  are  bent  into  a 
quadrant  of  a  circle.  An  efficiency  of  88  per  cent  has  been 
claimed  for  this  turbine  under  a  full  load. 

Comparison  of  Outward- flow  Turbines. — Fourneyron  deals 
with  a  varying  supply  of  water  by  means  of  a  circular  sluice, 
which  can  be  made  to  close  off  any  required  portion  of 
the  wheel.  A  similar  arrangement  may  be  added  to  the 
Cadiat  turbine,  which  is  of  the  outward-flow  type  and  is  fed 
from  above  through  a  cylindrical  reservoir,  the  upper  and 
lower  edges  of  the  reservoir  being  rounded  to  diminish  the  loss 
due  to  contraction.  The  objection  to  sluices  of  this  kind  is 
that  the  passages  no  longer  run  full  when  the  inlet  orifices  are 
partially  closed  and  there  is  therefore  a  considerable  diminution 
of  efficiency.  In  the  Whitelaw  turbine,  p.  375,  this  difficulty 
can  be  obviated  by  changing  the  outlet  instead  of  the  inlet  area. 

The  absence  of  guides  in  the  Cadiat  and  Whitelaw  turbines 
make  their  construction  somewhat  simpler,  but  their  efficiency 
is  comparatively  small,  that  of  the  Cadiat  being  about  65  per 
cent,  while  the  efficiency  of  the  Whitelaw  turbine  varies  from 
50  to  60  per  cent.  On  the  other  hand,  the  Fourneyron  turbine 
has  an  efficiency  of  more  than  70  per  cent  and  is  mechanically 
a  much  more  perfect  machine.  The  guides  in  the  turbine 
render  it  possible  to  utilize  almost  the  whole  of  the  energy  of 
the  water  either  by  equalizing  the  peripheral  and  relative 
speeds  at  the  outlet,  or  by  making  the  absolute  velocity  at  the 
outlet  radial.  The  Fourneyron  and  Cadiat  turbines  are 
specially  adapted  for  a  large  supply  of  water  and  a  moderate 
fall,  say  not  exceeding  about  30  ft.,  while  the  Whitelaw  tur- 
bines are  found  more  useful  for  a  small  supply  of  water  and  a 
high  fall. 


49$ 


THEORY  OF  TURBINES. 


FIG.  291.— Enlarged  Portion  of  Section  through  XV,  Fig.  287. 


FlG.  292. — Enlarged  Portion  of  Section  through  XY,  Fig.  285, 


-M,       m 


FlG.  293.— Enlarged  Portion  of  a  Cylindrical  Section  JTr,  Fig.  290, 
Developed  in  Plane  of  Paper. 


THEORY  OF  TURBINES.  497 

4.  Theory  of  Turbines  (Figs.  291,  292,  and  293). — Denote 
inward-flow,  outward-flow,  and  axial-flow  turbines  by  I.  F., 
O.  F.,  and  A.  F.,  respectively. 

Let  j\  ,  r2  be  the  radii  of  the  wheel  inlet-  and  outlet-surfaces 

of  an  I.  F.  or  O.  F, 

Let  r^ ,  r2  be  the  outer  and  inner  radii  of  the  wheel  inlet- 
surface  of  an  A.  F. 

/      r  _[__  r  \ 
Let  R   be   the   mean    radius    (= -)    of  an    A.    F., 

assumed  constant  throughout. 

Let  Al,  A2  be  the  areas  of  the  wheel  inlet-  and  outlet- 
orifices. 

Let  dl,  d2  be  the  depths  of  the  same  in  an  I.  F.  or  O.  F. 

Let  dv ,  d2  be  the  widths  of  the  same  in  an  A.  F. 

Let  //  be  the  depth  of  the  wheel  in  an  A.  F. 

Let  H l  be  the  effective  head  over  the  inlet-surface  of  the 
wheel.  This  is  the  total  head  over  the  inlet- 
surface  diminished  by  the  head  consumed  in 
frictional  resistance  in  the  supply-channel,  and 
by  the  head  lost  in  bends,  sudden  changes  of 
section,  etc. 

Then  H \  +  h  is  the  total  head  over  the  outlet  of  an  A.  F. 
available  for  work. 

Let  H 2  be  the  fall  from  the  outlet-surface  to  the  surface  of 
the  water  in  the  tail-race.  If  the  turbine  is 
submerged,  then  7/2  is  negative. 

Let  i\ ,  ?'2  be  the  absolute  velocities  of  the  water  at  the 
inlet-  and  outlet-surfaces. 

Let  z/j,  u2  be  the  absolute  velocities  of  the  inlet-  and  outlet- 
surfaces.  In  an  A.  F.  turbine  i^  —  u2. 

Let  Vl ,  V2  be  the  velocities  of  the  water  relatively  to  the 
wheel  at  the  inlet-  and  outlet-surfaces. 

Let  the  angular  velocity  of  the  wheel  =  &?  =  —  =  — . 

r\        rz 
Let  rf  designate  the  hydraulic  efficiency  of  the  turbine. 


498  THEORY  OF  TURBINES. 

Let  the  water  enter  the  wheel  in  the  direction  ac,  making 
an  angle  y  with  the  tangent  ad.  Take  ac  to  represent  v^ ,  and 
ad  to  represent  ur  Complete  the  parallelogram  bd.  The 
side  ab  represents  Vl ,  and  in  order  that  there  may  be  no  shock 
at  entrance,  ab  must  be  tangential  to  the  vane  at  a.  Again, 
at  f  draw  fg,  a  tangent  to  the  vane,  and//£,  a  tangent  to  the 
wheel's  periphery. 

Takey^-  and//£  to  represent  F2  and  u2  respectively.  Com- 
plete the  parallelogram  gk.  The  diagonal  fh  must  represent 
in  direction  and  magnitude  the  absolute  velocity  v2  with  which 
the  water  leaves  the  wheel.  Let  the  angle  hfk  =  #. 

Draw  cm  perpendicular  to  ad*  and  hn  perpendicular  to  gk. 

The  tangential  component,  viz.,  am  or  fn,  of  the  velocity 
of  the  water  as  it  enters  or  leaves  the  wheel  is  termed  velocity 
of  whirl  (v^). 

The  radial  component,  viz.,  cm  or  hn,  of  the  velocity  of 
the  water  as  it  enters  or  leaves  the  wheel  is  termed  velocity  of 
flow  (yr\ 

Take  vj  =  am,     vw"  —  fn, 

vrf  —  cm,      vr"  —  hn* 

Let  the  angle  bad  —  180°  —  a. 
Let  the  angle  gfk  —  180°  —  ft. 

Thus  a  and  ft  are  the  angles  which  the  vane  (or  blade)  tips 
(or  lips)  make  with  the  wheel's  peripheries. 
Then,  at  the  inlet-surface, 

•vw'  j=  i\  cos  y  —  accos  y  =  am  =  ad  ±  dm  =  ul —  Vl  cos  a,     (i ) 
vr'  =  i\  sin  y  =  cm  =  Vl  sin  a;     . (2) 

.and  at  the  outlet-surface, 

vw"  =  v2  cos  d  —  fn  —  fk  ±  kn  —  u2  —  Vz  cos  ft,      .     (3) 
vrff  =  v2  sin  d.  =  hn  =   F2  sin  ft .      (4) 


THEORY  OF  TURBINES.  499 

Let  Q  be  the  quantity  of  water  which  passes  per  second 
through  the  turbine.  Then,  disregarding  the  thickness  of  the 
vanes,  in  an  I.  F.  or  O.  F.  turbine 

V^  =  Vr'  -  271T&  =  Q  =  V#"  .  27rr2d2  =  Vr"A2  ,     .       (5) 
and  therefore 

'  ' 


Also,  if  d^  —  d2  —  d, 

' 


In  an  A.  F.  turbine 

vr/A1  =  vr'  .  27rR  .  d,  =  0  =  vr"  .  2^R  .  d2  =  vr/7A2,     (6) 
and  therefore 

" 


Q      =  v» 


Allowance  may  be  made  for  vane  thickness  as  follows : 
Let  6  be  the  angle  between  the  vane  of  thickness  BC  and 
the  wheel's  periphery  A B.    Then  the  space  occu- 
pied by  the  vane  along  the  wheel's  periphery  is 
AB=  BC  cosectf. 

Let  n  be  the  number  of  the  guide-vanes,  and  t 

their  thickness.  FIG.  294. 

Let  nl  be  the  number  of  the  wheel-vanes,  and  tir  t2  their 
thickness  at  the  inlet-  and  outlet-surfaces  respec- 
tively. 
Then,  in  a  radial-flow  turbine, 

Al  =  —^{2777^  —  nt  cosec  y  —  nltl  cosec  a}       .      (7) 


THEORY  OF   TURBINES. 
and 

-  «      cosec 


T9¥  being  a  fraction  depending  on  practical  considerations. 

In  an  axial-flow  turbine  R  is  to  be  substituted  for  rl  and  r2 
in  the  values  of  Al  and  A2. 

n^  may  be  made  equal  to  n  +  I  or  n  -f-  2. 

Work  and  Efficiency.  —  As  the  water  flows  through  the 
wheel,  let  v  be  the  velocity  of  flow  at  any  point  N  distant  r 


FIG.  295. 

(=  ON}  from  the  axis  O^  and  let  /  be  the  length  of  the  per- 
pendicular from  O  upon  the  direction  of  v.      Then 

v  —  momentum  of  moving  mass  of  water 

g 

=  impulse  on  wheel 

=  Fj  suppose. 
Therefore,  also, 

— vp  •=.  Fp  —  moment  of  couple  producing  rotation, 

e> 

and  the  useful  work  of  the  couple  per  second 

wQ 
•=.  rpoo  =  vpQD. 


THEORY  OF  TURBINES.  501 

But  if  vm  is  the  component  of  v  at  N  perpendicular  to  the 
radial  line  ON, 

vp 

vw  =  v  cos  B  =  —  , 

and  therefore  the  useful  work  of  the  couple  per  second 

wQ 
—W. 

Thus  in  an  I.  F.  or  O.  F.  turbine 

the  useful  effect  at  inlet    =  -^-voo      =  -^-v'u 


the  useful  effect  at  outlet  = 


o 


o  <5 

and  the  USEFUL  WORK  per  second  done  by  the  water  on  thw 
wheel  between  inlet  and  outlet 

wQ 

y  (Or,  -~  vw  'r2)c*,    .....      (9> 

wO 


The  EFFICIENCY  is  given  by  the  relation 

rj  X  wQH^  =  the  useful  work  per  sec. 
wQ 

~(^w\  -  VX)> 

or 

ijg^  =  VM,/UI  -  v^X  ,  .....     (ii] 

which  is  the  fundamental  equation  governing    the   design  of 
I.  F.  or  O.  F.  turbines. 
In  an  A,  F.  turbine 

wQ  wQ 

the  useful  effect  at  inlet     =  -  vJRoo  —  —  vju, 

o  o 

the  useful  effect  at  outlet  =  ^—v^'Roo  —  —  vjult 

o  o 


5°  2  THEORY  OF  TURBINES. 

and  the  USEFUL  WORK  per  second  done  by  the  water  on  the 
wheel  between  inlet  and  outlet 


w'  -  YW")R<»,      ........    (12) 

=    "JW  -  T."X  .......       ('3) 

The  efficiency  is  given  by  the  relation 

rj  X  wQ(Hl  -f-  //)  =  the  useful  work  per  sec. 


or 

^g(H1  +  h)-(v»/-vw")u1,    ....     (14) 

which   is  the  fundamental   equation  governing   the  design   of 
A.  F.  turbines. 

Again,   disregarding  hydraulic  resistances,  each  pound  of 

v  * 
water  on  leaving  the  turbine  carries  away  —  ft.-lbs.   of  energy. 

Hence 

the  USEFUL  WORK  in  an  I.  F.  or  O.  F.  turbine 


V  * 
the  corresponding  EFFICIENCY  being  I  --  ~,        .      .      (16) 

and  the  USEFUL  WORK  in  an  A.  F.  turbine 

1  +  h-      .....    (17) 


the  corresponding  EFFICIENCY  being  I  —  -  —  _  .  2      ...        (18) 

o  \      1      I          / 


THEORY  OF   TURBINES. 


Assuming  that  the  velocity  of  whirl  at  outlet,  viz. ,  vw" ',  is 
nil  and  that  H  is  the  portion  of  ffl,  or  of  Hl  -f-  h,  which  is 
transformed  into  useful  work,  then 

gH  —  u^J  =  ufa  —  vr'  cot  a), 
which  may  be  written  in  the  form 


I  = 


a  quadratic  giving 


«t  V     cot  or  //     z/,/    \2  cot2  a 

7w  ~~  '  ^Tff  ~^    V  ^  ~^W>  ~T~ 

This  result  has  been  employed  in  preparing  the  following 
Table  of  values  of  —  -1—   corresponding  to  different  values  of 


—  ^=  and  of  a  : 

VgH 


*•" 

15° 

30° 

45° 

60° 

75° 

90° 

105° 

120° 

135° 

150° 

165° 

1.0 

3.983 

2.189 

1.618 

•  329 

.142 

I 

.874 

•752 

.618 

•  457 

.251 

0.9 

3.629 

2.047 

1-547 

.188 

.128 

I 

.887 

•773 

•  647 

.488 

.271 

0.8 

3.289 

.909 

•477 

.090 

.114 

.900 

•795 

.677 

•524 

•304 

0.7 

2.952 

.776 

.409 

.225 

•095 

.908 

.819 

.709 

.563 

•339 

0.6 

2.621 

.647 

•344 

.188 

.083 

.923 

.842 

•  744 

.607 

.382 

0.5 

2.301 

.523 

.281 

•155 

.069 

•935 

.886 

.781 

•657 

-435 

0-45 

2.145 

•463 

.250 

•139 

.062 

.942 

.879 

.800 

.683 

.465 

0.4 

.991 

.405 

.220 

.122 

•055 

•949 

.890 

.820 

.712 

•499 

0-35 

.847 

•346 

.190 

.107 

.048 

•954 

.902 

.840 

-744 

•  541 

o-3 

.705 

•293 

.161 

.090 

.041 

.961 

.917 

.860 

•  773 

.586 

0.25 

.569 

.240 

.132 

.074 

•034 

.967 

•930 

.882 

.806 

.636 

0.2 

.440 

.188 

.105 

.059 

.027 

•973 

•943 

•90S 

.842 

.694 

0.15 

.318 

.138 

.078 

.044 

.O2O 

.980 

.966 

.927 

.878 

•758 

O.  IO 

.204 

.090 

.051 

.029 

.013 

.987 

.972 

•  951 

.917 

.830 

504  EXAMPLES. 

Allowance  may  be  made  for  the  principal  hydraulic  resist- 
ances by  taking 

z/2 
f2~l   as  the  loss  of  head  before  entering  the  wheel  and 

J72 

/4—  2   as  the  loss  of  head  before  entering  in  the  wheel-passages. 

<5 

Then  the  total  loss  of  head 

7/2  Y  2  v  2 

=/+/+-  .....  (.9) 


The  values  of  the  empirical  coefficients^  and/4  may  vary, 
the  former  from  .025  to  .20,  and  the  latter  from  .10  to  .20. 

Ex.  i.  Water  enters  an  O.  F.  turbine  of  3!  ft.  exterior  and  if  ft.  in- 
terior diameter  with  a  whirling  velocity  of  20  ft.  per  second,  and  leaves 
in  the  reverse  direction  with  a  whirling  velocity  of  10  ft.  per  second. 
The  wheel  makes  240  revolutions  per  minute.  Find  the  useful  head. 

it  .  1  1  .  240 
Ui  =  ---  =  22  ft.  per  sec., 

u%  =  2Ui  =  44  ft.  per  sec. 
Then,  if  H  is  the  useful  head, 

<wQH  =  work  done  in  driving  the  wheel 


O 

wQ  t  .    880 

=  —^-(22  x  20  —  44  x   (—  10)  )  =  wQ  .  -  , 

and  H  =  27*  ft. 

Ex.  2.  A  turbine  with  a  radial  inlet-lip  receives  10  cu.  ft.  of  water 
per  second  at  a  radius  of  2  ft.,  and  makes  105  revolutions  per  minute. 
The  water  enters  at  60°  with  the  wheel's  periphery,  and  leaves  without 
velocity  of  whirl.  If  the  efficiency  is  .88,  find  the  effective  head  and  the 
H.P.  of  the  turbine. 

Since  a  =  90°, 

105 
-  =  22  ft.  per  sec., 


60 


EXAMPLES.  505 

and 

.88  =  the  efficiency  =  -^  =  — l-f  =  — — -, 
gHi       32/fi      32//i 

or  Hi  =  17.1875  ft. 

The  H.P.  =  62?'10  x  MHi  =  17.1875. 

55° 

Ex.  3.  The  wheel  of  an  A.  F.  turbine  of  3  ft.  interior  diameter  has  a 
6-in.  width  of  orifice  opening  and  is  i  ft.  deep.  It  passes  33  cu.  ft.  of 
water  per  second  under  the  head  of  24  ft.  over  the  inlet,  and  the  water 
leaves  the  wheel  in  a  direction  given  by  cosec  8  =  1.015.  Determine  the 
efficiency. 

By  the  condition  of  continuity, 

TT     ,         2      „  t    „ 

or  vr'  =  6  ft.  per  sec. 

Therefore 

r/j  =  vr"  cosec  d  =  6  x  1.015  —  6.09  ft.  per  sec., 
and 

the  efficiency  =  i  -  o^\  t)   =  i  -  -^-  =  -9768. 


The  H.P.  =  x  25  x  .9768  =  2.929. 

32  55° 

Ex.  4.  Find  the  outlet  lip-angle  (/?)  from  the  following  data  :  radius 
to  inlet  =  twice  that  to  outlet  surface;  linear  speed  of  inlet  surface  = 
one-half  that  equivalent  to  the  effective  head;  inlet  velocity  of  flow  = 
one-eighth  of  that  equivalent  to  the  effective  head  ;  sectional  area  of 
waterway  is  constant  from  inlet  to  outlet  ;  the  water  leaves  without  ve- 
locity of  whirl. 


Then 


=  4     JJ\  =  2»,. 


By  condition  of  continuity, 

AiVr     =   A*Vr"    =   AiV 


and  v*  =  vr  = 


Hence  cot  ft  =  ^  =  —=^  =  2. 


506  EXAMPLES. 

Ex.  5.  The  wheel  of  a  turbine,  passing  10  cu.  ft.  of  water  per  second 
under  a  head  of  32  ft.,  is  6  ins.  deep  and  its  inlet-surface  has  a  diameter 
of  2  feet.  The  inlet-lip  is  radial  and  the  efficiency  may  be  assumed  to 
be  unity.  Find  the  guide-vane  lip-angle  and  the  power  of  the  turbine. 

* 

i  =  the  efficiency  = 


32  x  32       1024 
Therefore  Ui  =  32  ft.  per  sec.  =  v^' '. 

By  condition  of  continuity, 

It  .  2  .  $  .  Vr'  =   IO, 

or  vr'  =  —  =  3iaT  ft-  Per  sec' 

Hence  tan  y  =  —  =  —  =  .09943, 
and  y  =  5°  41'. 

The  H.  P.  =  62^  .  10 .  -^  =  36  fr. 

Ex.  6.  In  an  impulse  radial-flow  turbine  the  inlet-  and  outlet-orifice 
areas  are  equal  and  the  water  leaves  without  velocity  of  whirl.  Disre- 
garding hydraulic  resistances,  show  that  the  velocity  of  whirl  is  cos2  y. 

By  condition  of  continuity, 

Therefore  V*  =  v\  sin  y, 

and 

the  efficiency  =  i ^-  =  i  —  sina  y  =  cos2  y. 

Ex.  7.  In  a  radial-flow  impulse  turbine  the  peripheral  and  relative 
speeds  at  outlet  are  equal.  Show  that  the  direction  of  the  water  at 
inlet  bisects  the  angle  between  the  rim  and  the  inlet-lip.  Also  show 
that  rSdi  sin  2y  =  rjd*  sin  ft. 

But  Vi  =  #2 ,     and  therefore     Vi  =  uif 

so  that  2y  =  180°  —  a. 

By  condition  of  continuity, 

or  ridi  .  Fi  sin  a.  =  r^d*  .  V*  sin  ft, 

r<i 

or  r \d\u\  sin  2y  =  r*di  .u*  sin  ft  =  r*di  f  — «i  sin  ft. 

Or  rfdi  sin  2y  =  r-fd*  sin  ft. 


THEORY  OF   TURBINES.  507 

Application  of  Tor  rice  Hi  '  s  Principle.  —  If  —  ,  —  are  the 
pressure-heads  at  the  inlet-  and  outlet-surfaces  of  a  turbine 
wheel,  the  effective  head  over  the  inlet-orifices  is  H,  --  -  -  -. 


, 

iv 


Hence,  disregarding  hydraulic  resistances, 


•vr.  2  -n      _    •«% 

IN    A     REACTION    TURBINE    —  =  H,  —   -  -  -2.  (2O] 

2g  W 

In  turbines  of  the  impulse  type  pl  —  p2,  and  the  water  is 
usually  under  atmospheric  pressure  only  both  at  inlet  and 
outlet.  Thus 

V2 

IN     AN    IMPULSE    TURBINE       L  =  H,  .....  (2l) 

2g 

Allowance  may  be  made  for  the  loss  of  head  at  entrance 

into  the  wheel  by  substituting  —  r-  —  for  --  in  these  two  equa- 

5  c*  2g         2g 

tions,  the  average  value  of  the  empirical  coefficient  cv  being 
about  .949,  or  cz?  —  .9. 

Application  of  Bernouilli'  s  Principle. 

IN    A    REACTION    I.    F.    OR    O.    F.    TURBINE 

A  •   F*'  =  A  .   F22_^22-^2 

W  2g         W     '      2g  2g 

the  last  term  being  the  work  per  pound  of  water  due  to  centrif- 
ugal force.      Therefore 

V2  -     J72        7/2        *,  ^         /i         /, 

L2  __  ^  __  uz   —  u\    __  P\  —P*  x22x 

2g  2g  W 


or 

v2  V2—  V2        u2  —  u2 

1L   i  12  __  M.  .  .  Z?  __  ZL 

2g4  2g  2g 


5o8  THEORY  OF  TURBINES. 

which  may  also  be  written  in  the  form 

UlVl  COS  y          V22  —  U22     _ 

since,  from  the  triangle  acdy 

2,ujL\  cos  y  =  u*  +  v?  —  V?-       •      •      •      (25) 
IN    AN    IMPULSE    I.    F.    OR    O.    F.    TURBINE 

A  =  A »    and    v\  —  2gH\- 

Therefore  eq.  (23)  becomes 

772  __  772       2/2  _  u  2 

KZ  2       *  =     2  2       * (26) 

IN   AN   I.   F.    TURBINE  u,  >  «„  and  the  term  — is 

2<T 

negative.  Hence  eq.  (23)  shows  that  as  the  inlet  velocity  vl 
increases  or  diminishes  the  speed  of  the  turbine  diminishes  or 
increases,  and  that  therefore  the  centrifugal  force  tends  to 
maintain  a  steady  motion.  A  diminution  in  vl  also  necessarily 
leads  to  a  corresponding  diminution  in  the  loss  of  head  due  to 
hydraulic  resistances.  For  these  reasons  the  centrifugal  head 

u        f 
should  be  made  as  large  as  is  practicable,  and  the  ratio  —  =  — 

U2  r2 

is  usually  made  equal  to  2. 

IN  AN  O.    F.   TURBINE  u,  <  #„  and  the  term     2    ~ — l-  is 

g 

positive.  Hence  the  speed  of  the  turbine  increases  and 
diminishes  with  i\  ,  and  the  centrifugal  force  is  adverse  to  steady 
motion,  tending  both  to  augment  a  variation  from  the  normal 
speed  and  to  increase  frictional  losses  of  head.  The  centrif- 
ugal head  should  therefore  be  made  as  small  as  is  practicable, 

and  a  common  value  of  the  ratio  —  =  —  is  — . 

U2       r*      5 

IN   A   REACTION   A.    F.    TURBINE    each    fluid   particle    in 
passing  from  inlet  to  outlet  remains  at  the  same  distance  from 


THEORY  OF   TURBINES.  509 

the  axis,  and  therefore  no  work  is  done  by  centrifugal  force, 
but  an  additional  head,  //,  equal  to  the  depth  of  the  wheel,  is 
gained.  Then 

P      y2    A      v^ 

tV  2g         IV  2g 

or 

f  h  =  H^  +  //  -^  .       .     (27) 


2g  W 

Therefore 

v  2       V »  - 


2 


-^  =  ^-fh, (28) 


which  may  also  be  written  in  the  form 

ujv1^  +  V/i_u,  =  Hi  +  h)  ^ 

since  z^  =  ?/2. 

IN    AN    IMPULSE    A.    F.    TURBINE 

/i  =  /2  '       and      ^i2  :=  Z&^r 
Therefore  eq.  (28)  becomes 

VJ>  -  V  2 


In  order  to  secure  the  advantages  of  centrifugal  force, 
Belanger  proposed  that  the  wheel -passages  should  be  so  formed 
that  the  path  of  a  fluid  particle  would  gradually  approach  the 
axis  of  rotation. 

Lip  (or  Tip)  Angles. — The  angles  a  and  /3  which  the 
wheel-blade  tips  at  inlet  and  outlet  make  with  the  wheel's 
peripheries  are  generally  obtained  as  follows: 

From  the  triangle  acd, 
sin  (a  -j-  y)       ul 


=  — =  =  cos  y  -4-  cot  a  sin  y, 
sin  a  v,  r  " 


and  therefore 


u 

cot  (i  80°  —  a)  =  —  cot  a  =?  cot  y cosec  y.     (31) 


THEORY  OF   TURBINES. 


From  the  triangle  fkh, 
sin  (ft  +  tf) 


sn 


=  -i  =  -os  tf  +  cot  ft  sin 


and  therefore 


cot  (180°  —  ft)  =  —  cot  ft  =  cot  6 cosec  d.      (32) 

Conditions    Governing    the    Efficiency    of    Turbines.  —  The 
whole  of  the  water's  energy  should,  if  possible,  be  employed 
in  doing  useful  work  on  the  wheel,  and  the  water  should  there- 
fore  leave   the  'wheel   without  velocity,   or  vz  should   be   nil. 
This  condition  cannot  of  course  be  realized  in  practice,  as  no 
water  would  then  pass  through  the  wheel  and  consequently 
no  work  could  be  done.      For  purposes  of  efficiency  it  is  usual 
to  make  vz  small  by  adopting  one  of  the  following  hypotheses: 
EITHER  that  the  velocity  of  whirl  at  outlet  is  nil, 
OR  that  at  the  outlet  the  relative  velocity  of  the  water 
and  the  peripheral  linear  velocity  of  the  wheel  are 
equal. 

FIRST  consider  the  hypothesis    ' '  that  the  velocity  of  whirl 
at  outlet  is  nil."     Then 

(33) 


—  o. 


I.F. 


A.F. 


FIG.  296. 


FIG.  297. 


Thus    the   direction  of  v    is  radial  in  an  I.    F.    or  O. 


F. 


turbine,  Figs.  296  and  297,  and  vertical  in  an  A.  F.  turbine, 


THEORY  OF   TURBINES. 


Fig.  298,  and  therefore  the  angle  hfk  (=  6)  in  the  outlet  tri- 
angle of  velocities  must  be  a  right  angle.      Hence 


and 


z>2  =  vr"  =  u2  tan  ft  —  K2  sin  /J,     . 
772  —  7/2  —  7;  2  —  ?/  2  tan2  fl 

r   2  "9      " '  "     *  2      2       *'**•**      A^  *  •  • 

Also,  eq.  (5)  gives 

z/t  sin  yA^  =  v2A2  =  u2  tan  ftA2.    . 

General  Deductions . 


(34) 
(35) 

(36) 


IN  AN  I.  F.  OR  O.   F.  TURBINE 


Also,    disregarding    blade    thick- 
ness, 

A\  =  iTtr^di  and  y?a  =  27Cr*d*. 
Relation  between  the  lip-angles. 
By  eq.  (36)  and  the  triangle  acd> 


jdi  sin  y       u\       sin  (a.  +  y} 


*  tan  ft 


,     (37) 


or 


1     '  cot  ft  =  cot  y  +  cot  or.    .      (38) 


IN  AN  A.    F.  TURBINE 


Also,  disregarding    blade     thick- 
ness, 

A,  =  iitRdi  and  A*  —  2 


Relation  between  the  lip-angles. 
By  eq.  (36)  and  the  triangle  acd, 
d\  sin  y  _  Ui  _  sin  (a  +  y) 


2  tan 


sin  a 


or 


~-  cot  /Sf  =  cot  y  +  cot  a.     (38) 


REACTION    TURBINES. 


IN  AN  I.  F.  OR  O.   F.  TURBINE. 
Speed  of  turbine. 

By  eqs.  (24),  (35),  (37), 


IN  AN  A.  F.  TURBINE. 
Speed  of  turbine. 

By  eqs.  (28),  (35),  (37), 


tan  /?  +  2-7-  cot 


•      (39) 


Velocity  of  efflux. 


=  uJ  tan 


tan 


tan  /tf  +  2—  cot  X 


-.  .     (40) 


tan  /?  +  2-    cot 


(39) 


Velocity  of  efflux. 
vj  =  u<?  tan2  ft. 


tan  ft 


» 

tan   p  +  2~  cot 


(40) 


THEORY  OF   TURBINES. 


Amount  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 


i  tan  /? 


-.(40 


tan  ft  -f  2-f  cot  y 

The  useful  work    (disregarding   hy- 
draulic resistances) 


(42) 


(43) 


i  -\ — •  —  tan  ft  tan 
2  a% 

corresponding  efficiency 

__  i 

i  H —  -/  tan  ft  tan 


It  is  sometimes  assumed,  but, 
generally  speaking,  as  a  guide  only, 
that  the  inlet-lip  is  radial,  Figs.  299, 
300,  so  that 

a.  =  90°  and  u\  =  vw'. 


Amount  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 

Q  =   27tRdiVr"   =   2«AWiZ/a 


=2,M     /VV!i±*l£!Ll 

Y    tan  /?  +  2~  cot  y 


^<?   useful  work   (disregarding  hy- 
draulic resistances) 


I  +  —  -r  tan  ft  tan 

2     #2 

The  corresponding  efficiency 


(42) 


(43) 


It  is  sometimes  assumed,  but,  gen- 
erally speaking,  as  a  guide  only,  that 
the  inlet-lip  is  vertical,  Fig.  301,  so 


that 


a  —  90°  and  «i  =  vw'. 


Then 

UiVw  Wi1 

the  efficiency  =  —Jr  =  -77-.      (44) 


a      ttt    d 

--vrvci^ 


FIG.  300.  FIG.  301. 

Then 
the  efficiency  = 


h) 


,.     •     (44) 


THEORY  OF   TURBINES. 


An  approximate  estimate  of  the 
speed  of  the  turbine  may  now  be 
obtained  by  making  the  efficiency 
perfect,  when 

uJ=gHi..     .     .     (45) 

By  eqs.  (20),  (37),  (39),  the  difference 
between  the  inlet  and  outlet  pres- 
sure-heads 


\ridil 


K46) 


sinV(i-f  2-.-cot;Kcot/3)  I 
I  di  j 

If  the  turbine  is  above  the  surface 
of  the  tail-water,  there  will  be  no  in- 
flow of  air 

if  pi  >  pi,  i.e.,  if 

/         di  \       fr^\2 

sin'M  i +2 -cot  y  cot  fi  \  >  f  — —  1  . 

If  the  turbine  is  drowned  with  a 
head  h'  of  water  over  the  outlet, 
there  will  be  no  back-flow  of  water 

if  pi  >  pi  +  oo/i',  i.e.,  if 


sin2  y(i  +2-^  cot  y  cot  /3 


An  approximate  estimate  of  the 
speed  of  the  turbine  may  now  be 
obtained  by  making  the  efficiency 
perfect,  when 

u^=g(H,  +/;).       .     (45) 

By  eqs.  (20),  (37),  (39),  the  difference 
between  the  inlet  and  outlet  pres- 
sure-heads 

=       _tv 


W 


(46) 


I  \         «i  /J 

If  the  turbine  is  above  the  surface 
of  the  tail-water,  there  will  be  no  in- 
flow of  air 


if  pi  >  pi,  i.e.,  if 

<£, 
+  2~coty  cot/?^ 


& 


If  the  turbine  is  drowned  with  a 
head  h'  of  water  over  the  outlet, 
there  will  be  no  back-flow  of  water 

if  pi  >  pi  +  <*>h',   i.e.,  if 


sin2  r{  i  +  2-/-  cot  y  cot/5 


Speed  of  turbine. 
By  eqs.  (21),  (37), 

ri         ri  di  sin  y     .- 


IMPULSE    TURBINES. 

Speed  of  turbine. 
By  eqs.  (21),  (37), 


.  (47) 


Velocity  of  efflux. 
Vv=Uv  tan  /?  =  -^—  i  sin 


.   (48) 


Velocity  of  efflux. 

di    . 

V*  —  u*  tan  p  =  —  sin 
di 


.  (48) 


THEORY  OF   TURBINES. 


Quantity  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 

Q  =  2rtridiVr'  =  27tridiV1  sin  y 
=  2itridi  sin  y  V2gHi>     •     (49) 

The  useful  work  (disregarding  hy- 
draulic resistances) 


(50) 


(50 


r, 
=  coQffJi  - 


The  corresponding  efficiency 


Quantity  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 

Q  =  27tRdlvr'  =  iitRdwi  sin  y 
=  27tftdl  sin  y  ^gH^     .     (49) 

The  useful  work  (disregarding  hy- 
draulic resistances) 


The  corresponding  efficiency. 
H>      d?    . 


An  expression  can  also  be  easily  obtained  giving  the  effi- 
ciency (eq.  51)  of  the  A.  F.  turbine  independent  of  the 
head,  Hr  Thus,  by  eqs.  (28),  (35),  and  (47), 


^  sin  2y 
tan 


~" 


sin    y 
tan2 


h 


It  may  be  assumed,  as  a  first  approximation,  that  in  im- 
pulse turbines  the  whole  of  the  water's  energy  at  inlet  is 
transformed  into  useful  work.  Then 


Therefore 


=  2//T  COS  y  =  2 


COS 


SECOND.  Consider  the  hypothesis  "  that  at  the  outlet  the 
relative  velocity  of  the  water  and  the  peripheral  linear  velocity 
of  the  wheel  are  equal.  "  Then 


.(52) 


THEORY  OF   TURBINES. 


515 


The  triangle  of  velocities,  fkJi,    at  outlet    is  now  therefore 
an  isosceles  triangle,  in  which  fk  =  kh,  and  the  angle  hfk  =  $ 

=  90°  —  — .      Therefore 

v2  =  2«2  sin  -  =  2  F2  sin  - (53) 

Eq.  (5),  again,  gives 

Aj.\  sin  y  —  A2V2  sin  ft  =  A2u2  sin  ft.        .      .      (54) 


O.F. 


A.F. 


FIG.  302. 


FIG.  303. 


FIG.  304. 


General  Deductions. 

IN  AN   I.   F.   OR  O.   ¥.  TURBINE 

U\          Ui 
—   :=—=&?. 

Also,  disregarding  blade  thickness, 


Relation  between  the  lip  angles. 

By  eq.  (54)  and  the  triangle  acd, 
Figs.  302,  303, 
riVi  sin  y  _  «i  _  sin  (a  +  y) 


sin 


sin  a 


or 
-^rr 


cosec  ft  —  cot  x  +  cot  a.     (56) 


IN  AN  A.  F.  TURBINE 


Also,  disregarding  blade  thickness, 

Al  =  27TAW,  ;     *A*  =  27TAW2. 
Relation  between  the  lip  angles. 

By  eq.  (54)  and  the  triangle  acd, 
Fig-  304, 

di  sin  y  _  Ui       sin  (<x  +  y} 

di  sin  ft      z/,  ~  '  ' 


sin 


or 


^  , 

—  cosec  ft  =  cot  Y  +  cot  a.      (56), 


THEORY   Of-    TURBINES. 


REACTION    TURBINES. 


IN  AN  I.  F.  OR  O.  F.  TURBINE. 

Speed  of  turbine. 
By  eqs.  (24),  (52). 


and  hence,  by  eq.  (55), 

r22  </,  tan 


(57) 

,rQx 
(58) 


Velocity  of  efflux. 

-7/2    — 


ft 

sin2— 


i 
=  2P-//,-  tan  -  tan  y. 

aa  2 


(59) 


Quantity  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 

ft 

<Q  =  T.itr-id'i.Vr'  —  zrtrvdvVs  cos  — 


id^di  sin  ft  tan  y.    (60) 


The   useful  work  (disregarding   hy- 
draulic resistances] 


—  ^  tan  -  tan  y\     (61) 
The  corresponding  efficiency 

=  r  —  ^  fan  -  tan  y.    .     (62) 

di.  2 

By  eqs.  (20),  (57),  (58),  the  difference 
between  the  inlet  and  outlet  pres- 
sure-heads 


IV 


IN  AN  A.    F.  TURBINE. 

Speed  of  turbine. 
By  eqs.  (28),  (52), 

«i?/i  cos  r  =  £-(/fi 
and  hence,  by  eq.  (55), 


(57) 


Velocity  of  efflux, 
z/22=«22  sin2  — 


J  yO 

+  h)-±-  tan  -  tan  y.    (59) 


Quantity  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 


.  (60) 


The   useful  work  {disregarding   hy- 
draulic resistances] 


(7  sj  \ 

i  —  y  tan  -tan  y\  (61) 

7'^<?  corresponding  efficiency 

—  i  —  y  tan  --  tan  X-  .     (62) 

By  eqs.  (20),  (57),  (58),  /£*  difference 
between  the  inlet  and  outlet  pres- 
sure-heads 


4;  l 


THEORY  OF  TURBINES. 


517 


If  the  turbine  is  above  the  surface 
of  the  tail-water,  there  will  be  no  in- 
flow of  air 

if  p\  >  fa,  i.e.,  if 

sin  2Y       r<?di 
sin  ft       rfdi 

If  the  turbine  is  drowned  with  a 
head  h'  of  water  over  the  outlet, 
there  will  be  no  back-flow  of  water 

if  pi  >  pi  +  GO/I',  i.e.,  if 

H\  —  h'      r-fdi  sin  ft 
H       >r^~ 


If  the  turbine  is  above  the  surf-ice 
of  the  tail-water,  there  will  be  no  in- 
flow of  air 


,  i.e.,  if 
i  +  h  d* 


if  pi  > 

sin  2y 

sin  ~ft  > 


If  the  turbine  is  drowned  with  a 
head  h'  of  water  over  the  outlet*. 
there  will  be  no  back-flow  of  water 

if  /i  >  /2  -f-  fi»V,  i.e.,  if 


sin  iy      Hl  +  h 
> 


sin 


IMPULSE   TURBINES. 


IN  AN  I.   F.  OR  O.  F.  TURBINE. 

By  eqs.  (29),  (.52), 

«,  =  V\.    .     .     .     (64) 


IN  AN  A.    F.  TURBINE. 

By  eqs.  (30),  (52), 
igh  =  u<?  - 


=  uS  -  VS.    (64) 


FIG.  305. 

Then  the  inlet  triangle  of  veloc- 
ities acd,  as  well  as  the  outlet  tri- 
angle fkh,  is  also  an  isosceles  tri- 
angle, Figs.  305,  306,  and 

HI  =  da  =  dc  —   Vi. 
Therefore 

a  -  180*  -  2y.  .    .     (65) 


FIG.  306. 


Therefore 


=  V? 

=  «i2  -I- 


and 


—  2UiVi  COS  X, 


.   (65) 


THEORY   OF   TURBINES. 


Relation  between  the  lip  angles. 

By  eq.  (54)  and  the  isosceles  tri- 
angle acd, 

r*di  sin  y  _  u\  _  i 

^V2  sin"?  ~  ^  ~  ~2  S6C  Y' 


or 


sin  ft 


sin 


.     .     (66) 


Speed  of  turbine. 


VigHi.      (67) 


Velocity  of  efflux. 


sin  — 

ft  _  r* 2 

2  ~  t'i  cot  y 


(68) 


Quantity  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 

Q=2Ttri(fiVr'  =  2Ttrldrf^  sin  7- 

=  2itridiS\nytf2gHi,  (69) 

useful  work    {disregarding   hy- 
draulic resistances) 

- 


-        tan 


corresponding  efficiency 
3  — 


sn  — 


?  cosV 


=  i  — 7-  tan  y  tan  — . 
d*  2 


-J-     (70 

.     (72) 
(73) 


Relation  between  the  Up  angles. 
By  eqs.  (54),  (65), 
di  sin  y      uv       Hi  +  h  sec  y 


sin  fi 


or 


d*  Hi  +  h      sin  2y 
Speed  of  turbine. 


H\ 


""i          (66) 


//, 

^,+7^  sec  ^ 

//i  2 


T/, 


Velocity  of  efflux. 


//,  +  //     .    /^ 
— 77 —  sin  -  sec 

n.  i  2 


(67) 


i.  (68) 


Quantity  Q  of  water  passing  through 
the  turbine  per  second,  blade  thick- 
ness being  disregarded. 

sin  y 


.  (69) 

The   useful   work  (disregarding  hy- 
draulic resistances) 


corresponding  efficiency 


_-.  j  yy 

=  I 77T~~  si"2  "  sec2  r      (72) 

=  i  -  -j  tan  r  tan  — .       .     .     (73) 


PRACTICAL    COEFFICIENTS.  519 

5.  Remarks  on  the    Efficiency. — The  expressions  giving 
the  efficiency  in  the  preceding  deductions  are  all  independent 
•of  the    head,  and    it  follows  that  turbines  work   equally  well 
•above  and  below  water. 

The  efficiency,  again,  increases  as  the  ratio  ~  diminishes, 

"but  it  should  be  remembered  that  the  value  of  dv  must  not  be 
too  small,  as  this  might  cause  a  contraction  at  entrance  and  a 
corresponding  loss  of  energy.  The  wheel-passages  should 
always  run  full  bore,  and  therefore  d^  must  not  be  too  large. 

Finally,   the    efficiency   increases    as  the   angles  ft  and    y 
'diminish. 

6.  Practical  Values. — The  following  are  the  values  which 
experience  indicates  as  giving  good  results  in  practice,  but  they 
should  be  only  regarded  as  guides : 

Let  i'  be  the  theoretical  velocity  due  to  the  head  H^  so  that 
z>~  =  2gHr     Then 

In  an  /.  F.  reaction  turbine 


^=•4  =  -'. 

2  <•> 


y  usually  varies  from  10°  to  30°,  an  average  value  being 
20°. 

If  u  =r  K  ft  usually  varies  from  135°  to  150°,  an  average 
value  being  145°. 

If  v^'  =  o,  ft  usually  varies  from  30°  to  45°,  an  average 
value  being  35°. 


520  PRACTICAL    COEFFICIENTS. 

In  an  O.  F.  reaction  turbine 

v 


Vr"  =    .21V  tO   .17^, 


Let  #  be  the  number  of  the  guide-blades. 
Let  //j  be  the  number  of  the  wheel-blades. 
Then 


=  4  X  shortest  distance  between  wheel-blades, 

2,r 
-  =  shortest  distance  between  guide-blades, 

n  =  -;/!  to  -»r 


The  H.  P.  =  .ijrfH. 

y  usually  varies   from  20°  to  50°,  an   average  value  being- 

25°. 

ft  usually  varies  from   20°  to   30°,  an  average  value  being; 

25°- 

In  an  A  .  F.  reaction  turbine  % 

»/=  vr"  =-i$v  to  ,2v, 
u^  —  u.2  =  .567-. 

Y  usually  varies  from    15°  to   50°,  an  average  value  being 

25°. 

.ft  usually  varies  from    12°  to   30°,  an  average  value  being 

25°. 

For  a  delivery  of  30  to  60  cu.  ft.  and  a  fall  of  25  to  40  ft. 

r=i5°toi8°     and     ft  =  13°  to  16°. 
For  a  delivery  of  40  to  200  cu.  ft.  and  a  fall  of  5  to  30  ft. 

r  =  1  8°  to  24°     and     ft  =  16°  to  24°. 


EXAMPLES.  521 

For  a  delivery  of  more  than  200  cu.  ft.  and  for  falls  of  less 
than  about  5  or  6  ft. 

y  =  24°  to  30°     and      fi  =  24°  to  28°. 
Denoting  \/Al  sin  y  by  A', 

R  may  vary  from  -A'  to  2 A'  if  A'  <  2  sq.  ft. 


2 
"     "        "        "     -Af  to- A  if  A'  >  2  sq.  ft.  and  <  16  sq.  ft. 

i  *"* 

"     "        "        "      ^r  to  -^'  if  A'  >  200  sq.  ft. 

4 

In  A.  F.  impulse  turbines  R  is  often   made  to  vary  from 

-A'  to  2^4 '. 
4 

In  reaction  and  impulse  turbines  the  blade  thickness  varies 
from  -J  to  f  in.  if  the  blades  are  of  wrought  iron,  and  from  %  to 
-f  in.  if  they  are  of  cast  iron.  The  tips  of  cast-iron  blades  are 
usually  tapered. 

Ex.  i.  An  axial-flow  impulse  turbine  passes  170  cu.  ft.  of  water  per 
second  under  the  head  of  8.6  ft.  over  the  inlet,  and  it  may  be  assumed 
that  the  whole  of  this  head  is  transformed  into  useful  work.  The  depth 
of  the  wheel  is  .9  ft.,  its  mean  diameter  is  8.4  ft.,  and  the  outlet-lip  makes 
an  angle  of  72°  with  the  vertical.  The  turbine  has  62  guide-  and  60  wheel- 
vanes,  all  the  vanes  being  ^  in.  thick.  The  outlet  velocity  of  whirl  is 
nil.  Find  the  direction  of  motion  of  the  water  at  inlet,  the  slope  of  the 
•wheel  vane  at  inlet,  the  H.P. ,  the  speed,  and  the  inlet  and  outlet  orifice 
areas  and  widths. 

First.     Disregard  hydraulic  resistances. 

Z/,a  //,VW'  UiV-    COS  Y 

Then  --  =  8.6  =  — —  —  -    — - — -, 

and  Vi  =  '21/1  cos  y  =  8  |/8.6 

=  23.4606  ft.  per  sec. 

AISO,  I7,2   =   V-?  -f  «ia   —   27/i//i   COS  y   =  11^   =  7/2a. 

Therefore     V\  —  u^  =  «a,  and  the  triangle  acd  is  isosceles,  so  that 
a  —  1 80°  —  iy. 

9  f     2  Q   /C 

Again, =  the  efficiency  =  i =    '     =  .905, 

2^  x  9.5  2g       9.5 

and  v-i  =  8  \/.$  =  7.58946  ft.  per  sec. 


522  EXAMPLES. 

Therefore          u,  =  u*  =  v,  cot  18°  =  23.358  ft.  per  sec., 

2W, 

and  sec  y  =  —  =  1.99125, 

so  that  y  —  59°  52', 

and  <*  =  1 80°  —  2  y  —  60°  16'. 

The  H.P.  =  6*JL!Z?JL*5  x  i905  =  166.,36. 

60  x  23.358 
The  speed  in  revolutions  per  mm.  =  — g =  53.08. 

The  inlet  area     =  ^V  = ^—  =  8.38  sq.  ft. 

z>r         ^j  sin  y 

The  outlet  area  =  —  =  -^£-  =  22.4  sq.  ft. 
^        7-589 

8.38  =  di  j  TT  x  8.4 cosec  59°  52' cosec  60°  16'  >  =•  tt,  x  20.53396. 

and  dt  =  .408  ft. 

22.4  =  d*  <  Tt  x  8.4 cosec  1 8°  \  ~  dt  x   18.30983, 

(  24  )  : 

and  </2  =  1.223  ft» 

Second.     Take  the  hydraulic  resistances  into  consideration. 

7/:Q  _    8  «-.Vw    _  «i7/i  COS  y 

^  ~~  ~9(  '  g  g 

Therefore          v    =  2«,  cos  y  —  22.1189  ft.  per  sec.          , 
The  triangle  acd  is  theretore  isosceles,  and 

V\  =  «.  =  wa, 

so  that  nr  =  i8oe  —  2  y. 

Also, 

~  F,'  =  ^»,f  sec5  r  =  F,3  -h  2^/^  =  »,«  +  57.6  =  ^«i«  sec1  r. 

Therefore     '  «  '(i.i  sec'  y  -  \ )  =  57.6, 

and  w    =  16.325  ft.  per  sec.  =  w«. 

z/.        22.1180 
Then  cos^  =  _=  =  ___  =  .677453. 

and  y  —  47°  21'. 

Hence,  too.  «:  =  180°  —  2  ^  =  85°  18 . 

6c  x   16.325 

The  speed  in  revolutions  per  mm.  = =37.1. 

ft  x  0.4 

The  efficiency  =  - —    =  .8046. 


EXAMPLES.  523 


The  H.P.  ="       -^      ?  x  .8046  =  147.676. 

170          170  cosec  47°  21' 
The  inlet  area    =  —-fa-j.  =     22  Ilg =  i<>-449  sq.  ft. 

1 70  1 70 

The  outlet  area  =  —  = 5  =  32.05  sq.  ft. 

v-i        u-i  tan  lo 

9  ,  (                      62                               60  n,  ) 
10.449  =      **'  )  n  x  8.4 cosec  47°  21' cosec  85    18  > 

10  '  24  24  ) 

=  18.3402  x  di, 
and  dt  —  .57  ft. 

9  ,  \  60  ) 

32.05  =  —  -a*  -}  TC  x  8.4 cosec  18°  r 

J       10      (  24  ' 

=  d't  x  14.85885, 
and  r/2  =  2.157  ft. 

Ex.  2.  An  A.  F.  reaction  turbine  of  7  ft.  mean  diameter  passes  198 
cu.  ft.  of  water  per  second  under  a  total  head  of  13.5  ft.,  the  depth  of  the 
wheel  being  i  ft.  At  inlet  the  lip  angle  (a)  is  90°,  and  at  outlet  the 
peripheral  and  relative  velocities  are  equal  ( V*  =  u*  =  «i).  The  width 
of  the  wheel  is  i  ft.  at  inlet  and  1.25  ft.  at  outlet.  Determine  the  di- 
rection and  magnitude  of  the  velocity  of  the  water  at  entrance,  the  up 
angle  at  outlet,  the  speed  in  revolutions  per  minute,  the  efficiency  and 
the  H  P.  Disregard  hydraulic  resistances. 

By  the  condition  of  continuity, 

7T  .  7  .    I   .  Vr     =    198    =    7C  .   7  .    ityr", 

and  therefore  vr'  =  9  ft.  per  sec.,  vr"  =  j\  ft.  per  sec. 

Again, 

64    X    13.5    -  7',2  =   864  —  Vr'9  -    «i*   =     JV   —     VS   =   Ui*   -  Vr'  \ 

or  2«i*  =  864,     or     MI  —  12  4/3  ft.  per  sec.  —  M9, 

MI  12  4/7 

cot  y  —  — ;  =  -    — -  =  2.309,     and     y  =  23°  25  , 

Vr  9 

sin  ft  =  y-  =  ~  =  -1^  --  =  -  4/3  =  .3464,     and     ft  =  20°  5'. 
Therefore  8  =  ^(180°  -  20°  5')  -.=  79°  77^', 

v*  =  2wa  sin  —  =  24  4/3  x  .1744  =  7.25  ft   per  sec. 

The  efficiency  =  i  -  z — — =  i  -  .0608  =  .9391. 

64  x   13.5 

The  H.P.          =  62*  X  T?:!  X   I98    x  .9391  =  285.25, 


524  EXAMPLES. 

60  x  1  2  f/3 
Revolutions  per  min.  =  —  —    =  56.68. 

7t     X     / 

Ex.   3.    To  construct    an   O.  F.   turbine   from   the  following  data  ; 
the  fall  (//i)=  5  ft.  ;  the  interior  diameter  (2^1)  =  1.8  ft.;  the  exterior 
diameter  (2r2)  =  2.45  ft.  ;   Q  =  30  cu.  ft.  per  second;  y  —  30°  ;  the  effi- 
ciency (rf)  =  .9.     Also,  disregard  hydraulic  resistances. 
First.     Take  vw"  =  o.     Then 

v<?      _  UiV\  cos  300 
~64Tx~s=~       32  x  5 
Therefore  v<>  =  4  ^2  ft.  per  sec., 

and  UiVi  =  96  \  3. 

Again,  by  the  condition  of  continuity  (eq.  5), 

TT  x    i.  80  x  d\v\  sin  30°  =  30  =  n  x  2.45  x  </ava. 
Taking  d\  =  d*  , 

.yvi  —  2.457/2  —  9.8  4/2, 

98    /— 
and  Vi  =  —  V  2  ft.  per  sec. 

Therefore  Ul  =  ^^  =  1^  ^/- 

9b      ^       49 


and  Ua  -  ?^«,  =  6  V6  ft.  per  sec. 

I  .o 

Hence       si"  (g  +  -^  =  *'  =  2L*S  =  tl  +  L  cot  «. 
sin  a  7/1          2401  2         2 

or  cot  (1800  —  a)  =  1/3  .  -   _  =".32966, 

and  a°  =  108°  15'  =  inlet-tip  angle. 

Also,  tan  ft  =  -  =  ±-^i  =  .3849, 

»»       6  4/6 

and  ft  —  21°  3'  =  outlet-tip  angle. 

Disregarding  the  thickness  of  the  vanes, 

the  inlet  area    =  Ai  =  --,  =  --  ^—~  =  —  4/2  =  3.8963  sq.  ft., 
'  sin  30°        49  J 

and  y,  -       ~       -  .6886  ft.  =  &; 


the  outlet  area  =  A*  =  —  =        -  =  5.303  sq.  ft. 

4 


EXAMPLES.  525 


The  number'of  revolutions  per  min.  =  -^-^     .  ^  . 


^         Second     Take  V,  =  u,  =  ^ut.     Then  the  triangle/^  is  isosceles. 


Therefore  z/2  =  2#2  sin  -  =  vw"  cosec  -, 

and  2/2  2  =  2u?v  " 

Again,  i  _  -^—  =  .9  = 


64  x  5  32  x  5 

cos  30° 


1 60 

Therefore      v9  =  4  4/2"  ft.  per  sec., 

UlVl  =  ¥?  i/~=v2~Su 
3  Vl  1.8  "2 


cosec  -. 


2 


By  the  condition  of  continuity, 

22  22  22  /J 

-rfiVi  sin  30°  x  1.8  =  30.=  -d&r"  x  2.45      =  ~-d&t  cos  -   x  2.45 

22   ,  >-  /f 

=  —  «a  9.8  r  2   COS  -. 
7  2 

Taking  ^  =  ^  t 

.9^/i  =  9.8  i^2  cos  -. 
Hence  1^  4/3"  =  9-8  ^2  CQS  /? 

3  n  2 

or.  cot  |  =  432Q  j/3   =  3 

anc*  ^  =  35°  34'  =  outlet-tip  angle. 

Hencet  also 

Vi  =  14-6634  ft.  per  sec.,  ^2=  9.261  ft.  per  sec.,  and  //,  =  6.7963  ft.  per  sec 

j        Again,     8in  f«  +jg)  ^  «»  =   1.8    ^         '•«.      ^         ^9 

sm  a  2/j       2.45  zv       2.45  /tf'  x_         ^» 

or          cos  30°  +  cot  a  sin  30°  =  - — ^ —  cosec  /J  =  .46399, 

or  cot  (180°  —  a)  =  .804, 

*nd  «  =  128°  48'  -  inlet-tip  angle. 


526  EXAMPLES. 

Disregarding  the  thickness  of  the  vanes, 

30  30  54  /3 

the  inlet  area    =  A,  =  —  ,  =  -  :  -  -  =  .-  sec  —  =  4.092  sq.  ft.,. 

Vr          Vi  Sin   30°         9.8  4/2  2 

4  092 
and  d,  =  n  x  i  8  =  .7233  ft.  -  d*  ; 

the  outlet  area  =  —  ^-—  a  =  —  -^rr  x   1.05018  =  5.5694  sq.  ft. 
t*  cos  I      2  ^ 

60  x  6.796 
The  number  of  revolutions  per  mm.  =  -  —  =  72. 

Ex.  4.  An  I.  F.  reaction  turbine  of  24  ins.  exterior  and  12  ins.  interior 
diameter  passes  400  gallons  of  water  per  second.  The  inlet  and  outlet 
orifice  areas  are  equal  and  the  depth  of  the  latter  is  1.25  ft.  The  guide- 
vane  lip  has  a  slope  of  i  in  5  and  the  inlet-lip  is  radial.  Disregarding 
vane  thickness  and  hydraulic  resistances,  find  the  total  head  over  the 
inlet  and  also  the  efficiency,  the  outlet  velocity  of  whirl  being  nil. 

By  the  condition  of  continuity, 

AiVr'   =  AiVr     =  4OO  -3-  6i  =  64  =  A  Mr"   =  AiVt. 

Therefore 

Vr'-  =  Vr"   =  V*   =  64  -5-   7t  .   I   .   I±   =   8^5  ft.  per  SCC., 

and  the  head  equivalent  to  z/a  =  p-  =  f  —  J  =  1.036694  ft. 

£  \jj/ 

Again, 

Ui  =  Vr  cot  y  =  $vr'  =  4oT87  ft.  per  sec.  =  2«3, 

and  the  useful  head  =  U^-  =  —  =  (4OT8T>  -s-  32  =  51^  ft. 

*  o 

Hence 

the  /0/d/head    =  1.036694  +  51.834710  =  52.871404  ft., 


and  the  efficiency  =  =  .98. 

7       52.871404        y 


Also,  the  speed  in  revolutions  per  min.  =  -  X  4C>TT  =  388.76. 
The  H.P.  = 


It    X    2 

62*  X  64  x 


55° 


^ 
tan  /?  =  —  =  8F85  -5-  2oT4r  =  .4,     and     /5  21°  48'. 


EXAMPLES.  527 

Ex.  5.  In  the  preceding  example  show  how  the  results  will  be  mod- 
ified if,  instead  of  the  outlet  velocity  of  whirl  being  nil,  the  relative  and 
peripheral  velocities  at  outlet  are  equal. 

As  before, 

vr'  =  vr"=  858-  ft.  per  sec., 

ft.  per  sec.  —  2«2  =  2  1/2". 


The  speed  in  revolutions  per  min.  =  -  U  =  388.76. 

/t    X    2 

Again,  V*  -  V?  =  «,«  -  «,*  + 

Or  «2J   —  2/r'  a  =   Wa'  —   «j*    + 

and  H,  ,  the  total  head,  =  — 

o 


Also, 


sin  0  =  -r-  =     -  =  8&  -8-  20T4T  =  .4,     and     ft  =  230  35'. 


The  effiaency  -,_ 


''  ^    = 


32  x  51.834710 
The  H.P.  =  ^£_i^r2_i^i!r^ii  x  .979  =  369.109- 

Ex.  6.  Avortex  impulse  turbine,  without  guide-vanes  but  with  32  wheel- 
vanes  of  f-in.  thickness,  has  an  exterior  diameter  of  2.625  ft.,  an  interior 
diameter  of  2.1  ft.,  and  passes  30  cu.  ft.  of  water  per  second  under  a  head 
of  560  ft.  The  water  enters  at  an  angle  of  30°  with  the  wheel's  periph- 
ery, and  the  relative  and  peripheral  velocities  at  outlet  are  equal.  The 
wheel  depth  at  outlet  is  3  times  the  depth  at  inlet.  Allowance  is  made 
for  hydraulic  resistances  by  taking  .94  as  a  coefficient  of  velocity  at  inlet, 
and  by  adding  10  per  cent  to  the  head  equivalent  to  the  relative  velocity 
at  outlet. 

T/J  =  .94  1/64.560  =  177-955  ft.  per  sec. 

1 1 

Also,  U-?  —  u\    —  —  Fy 

10 

«  _        a  }       i   — 

1—1         10 


.  /25o         HI        sin  (<x 
Therefore  V  tt=  "//.   =  ~ 


Vi  sin  30° 


528  EXAMPLES. 


or  sin  (a  +  30) 


=  -^  \  =  48473. 


«  +  30°  =  151°,     and     «  =  121°. 
sin  (a  +  30°)  sin  29° 


—  100.634  ft.  per  sec., 


Again,  „,=„.__    _          =  ,77.955 

— 
and  u*  =  80.5072  ft.  per  sec. 

60  x   100.634 
The  speed  in  revolutions  per  min.  =  --------  -^-  —  =  731.08. 

By  the  condition  of  continuity, 

AiVr  =  AiVi  sin  y  =  30  =  A*vr"  =  A^Vi  sin  ft  =  A*u*  sin  fi. 
Therefore 


x  2.625  —  32  x  ~Q(=^>~  —  cosec  30°= 

=  -337941  sq.  ft., 
and  di  =  .06  ft.  =  — . 

Also, 

.  (  1  ,  )  30 

cosec 


10 


.I8-J7TX2.I—  32  x—  -  cosec  ft  [  —  Ai  =  — 
(  48  |  », 


3°  /? 

—  -  —      -  cosec  p, 
80.5072 

or  1.0692  =  cosec  fi  (.324  +  .372637)  =  cosec  ft  x  .696637, 

and  cosec  ft  =  1.534,     or    ft  =  40°  41'. 

Therefore,  also,  8  =  69°  39^'. 

Again,        vw'  =  177.955  cos  3°°  =  I54-II3  ^.  Per  sec., 
and 

W  =  «2(i  —  cos  ft)  =  80.5072  x  .241676  —  19.4723  ft.  per  sec. 

Hence 

the  efficiency  =  loa6*  X   154"'  '3  "  8°'5°72  X   '»4y'} 

32  x  560 

=  '3941  .?3 
17920 


The  H.P.  = 


DRAFT-TUBES.  529 

7.  Theory  of  the  Suction  (or  Draft)  Tube.  —  Vortex  and 
axial-flow  turbines  sometimes  have  their  outlet-orifices  opening 
into  a  suction  (or  draft)  tube  which  extends  downwards  and 
discharges  below  the  surface  of  the  tail-water.  By  such  an 
arrangement  the  turbine  can  be  placed  at  any  convenient  height 
above  the  tail-  water  and  thus  becomes  easily  accessible,  while 
at  the  same  time  a  shorter  length  of  shafting  will  suffice.  The 
suction  tube  is  usually  cylindrical  and  of  constant  diameter,  so 
that  there  is  an  abrupt  change  of  section  at  the  outlet-surface 
of  the  turbine,  producing  a  corresponding  loss  of  energy  by 
eddies,  etc.  This  loss  may  be  prevented  by  so  forming  the 
tube  at  the  upper  end  that  there  is  no  abrupt  change  of  section, 
and  by  gradually  increasing  the  diameter  downwards.  The 
cost  of  construction  is  greater,  but  the  action  of  the  tube  is 
much  improved. 

Let  h'  be  the  head  above  the  inlet-orifices  of  the  wheel. 

Let  h"  be  the  head  between  the  inlet-orifices  and  the  sur- 
face of  the  tail-water. 

Let  L^  be  the  loss  of  head  up  to  the  inlet-surface. 

Let  L2  be  the  loss  of  head  between  the  wheel  and  the  tube- 
outlet. 

Let  z/4  be  the  velocity  of  discharge  from  the  outlet  at  bottom 
of  tube. 

Let  P  be  the  atmospheric  pressure. 

Then,  assuming  that  there  is  no  sudden  change  of  section 
at  the  outlet-surface, 


W 

and  therefore 


P  V  2  V  2 

£JL  _l_  12   _   1  4     I 

W   ^          ~  ~^ 


53°  DRAFT-TUBES. 

where  //  =  h'  -\~  h"  =  total   head    above    tail-water    surface, 
and  7'22,  v±,  Ll ,  L2  are  expressed  in  the  forms 


<  I 


,  /*4  ,  /i5  ,  //6  being  empirical  coefficients. 
Again,  the  effective  head 


.  2 


and  is  entirely  independent  of  the  position  of  the  turbine  in  the 
tube. 

Also,  if  A  i  is  the  area  of  the  outlet  from  the  suction-tube, 

^474=  Q  ~  A<i\  sin  Yi 

so  that  7'   can  be  expressed  in  terms  of  v.,  and  hence  —  -  2 

ze/ 

is  also  independent  of  the  position  of  the  turbine  in  the  tube. 

Suppose  the  velocity  of  flow  to  be  so  small  that  z/4,  v2,  L.2 
may  be  each  taken  equal  to  nil.      Then 

*"+*  =  £; 

W  W 

and  since  the  minimum  value  of  p2  is  also  nil,  the  maximum- 
theoretical  height  of  the  wheel  above  the  tail-water  surface  is 
equal  to  the  .head  due  to  one  atmosphere.  Again, 

g(h>  +  //')  =  gH  =--  r.X  -  Vw"u,+  V-± 

v  2 
=  ^  cos  yUl  -  u2(u2  —  V2  cos  fl)  +  -£. 

But 

AJL\  sin  7  —  Q  =  A2v2  sin  6  =  A2V2  sin  /?  =  A±v±\ 
and  hence,  taking 


cos  .  u   cos   ?   —  «2       -- 


LOSSES  AND  MECHANICAL  EFFECT.  53* 

and  therefore 


—  cos  y  4-  v  w0 .  cos 


where  B  =  —  f  —  cos  y  -\-  y//3  cos  p\ 


Hence  it  follows  that  vl  increases  with  u2,  i.e.,  with  the 
speed  of  the  turbine,  if 


A  suction-tube  is  not  used  with  an  outward-flow  turbine, 
but  a  similar  result  is  obtained  by  adding  a  surrounding  sta- 
tionary casing  with  bell-mouth  outlet.  A  similar  diffusor 
might  be  added  with  effect  to  a  Jonval  turbine  working  without 
a  suction-tube  below  the  tail-water.  The  theory  of  the 
diffusor  is  similar  to  that  of  the  suction-tube. 

8.  Losses  and  Mechanical  Effect. — The  losses  may  be 
enumerated  as  follows: 

I.  The  loss  (Z,j)  of  head  in  the  channel  by  which  the  water 
is  taken  to  the  turbine. 

L     -f1-^ 
1~/1  m2g' 

_fl  being  the  coefficient  of  friction  with  an  average  value  of 
.0067,  /  the  length  of  the  channel  of  approach,  m  its  mean 
hydraulic  depth,  and  z'0  the  mean  velocity  in  the  channel. 

Ll  is  generally  inappreciable  in  the  case  of  turbines  of  the 
inward-  and  axial-flow  types,  as  they  are  usually  supplied  with 
water  from  a  large  reservoir  in  which  ?'0  is  sensibly  nil. 

If  AQ  is  the  sectional  area  of  the  supply-channel,  then 

-<Vo=  (2=  ^1^1  sin  7, 


S32  LOSSES  AND  MECHANICAL   EFFECT. 

and 


A, 

31.   The  loss  (Z2)  of  head  in  the  guide-passages. 
This  loss  is  made  up  of: 

(a)  The   loss   due   to   resistance   at  the  entrance  into   the 
passages  ; 

(b)  The  loss  due  to  the  friction  between  the  fluid  and  the 
fixed  blades; 

(V)   The  loss  due  to  the  curvature  of  the  blades  ; 
(d)  The  loss  of  head  on  leaving  the  guide-passages. 
These  four  losses  may  be  included  in  the  expression 


f2  being  a  coefficient  which  has  been  found  to  vary  from  .025 
to  .2  and  upwards.  An  average  value  of/2  is  .125,  but  this  is 
somewhat  high  for  good  turbines. 

NOTE.  —  In  impulse  turbines  f2  has  been  found  to  vary  from 
.11  to  .17. 

III.   The  loss  (Z3)  due  to  shock  at  entrance  into  the  wheel. 

c,   In  order  that  there  may  be  no  shock 

'"'/I     at  entrance,   the  relative  velocity 


yj   |     must   be   tangential   to   the    lip   of   the 
vane.     For  any  other  velocity  (z\'  =  ac') 


(X\\  , 

and  direction  (dac  —  y')  of  the  water 
FIG.  307.  at  entrance,  evidently 

(ccj       (c'of  4-  (coj 
=  the  loss  of  head  =  V—  •+-  =  ^     1-J^^L 

(y'  sin  y\—  v^  sin  ^)2        (z/  cos  y'  —  v^  cos  y)2 


(V  sin  yf  —  Vl  sin  a)2       (vf  cos  y'  -  -  vl  —  Vl  cos  <xf 


LOSSES  AND  MECHANICAL  EFFECT.  533 

Generally  co  is  small,  and  Z3  is  always  nil  when  the  turbine 
is  working  at  full  pressure  and  at  the  normal  speed. 

This  loss  of  head  in  shock  caused  by  abrupt  changes  of 
section,  and  also  at  an  angle,  may  be  avoided  by  causing  the 
section  to  vary  gradually,  and  by  substituting  a  continuous 
curve  for  the  angle. 

IV.  The  loss  (Z,4)  of  head  due  to  friction,  etc.,  in  passing 
through  the  wheel-passages,  including  the  loss  due  to  leakage 
in  the  space  between  the  guides  and  the  inlet-surface.  This 
loss  may  be  expressed  in  the  form 


T 

~ 


4  ~4  2g  ~42  sin 


Aicosr 


\*vi\ 

!       > 


where /4  varies  from  .  10  to  .20. 

NOTE. — The  loss  of  head  due  to  skin-friction  often  governs 
the  dimensions  of  a  turbine,  and  renders  it  advisable,  in  the 
case  of  high  falls,  to  employ  small  high-speed  turbines. 

V.  The  loss  of  head  (Z5)  due  to  the  abrupt  change  of  sec- 
tion between  the  outlet-surface  and  the  suction-tube. 

As  in  III,  v2(=f/i)  is  suddenly  changed  into  v%  (=  fh')> 
and  the  loss  of  head  is 


2g  2g 


since  h  '  x  is  very  small  and  may  be  disre-         Uz 
garded.     Thus  FlG-  3<*. 


*^    — 


vj  being  the  component  of  v^  (fhf)  in  the  direction  of  the  axis 
of  the  suction-tube. 

If  there  is  no  abrupt  change  of  section  between  the  outlet- 
surface  and  the  tube,  Z5  is  nil. 

VI.  The  loss  of  head  (Z,6)  due  to  friction  in  the  suction- 
tube.  Assume  that  the  velocity  v^  of  flow  in  the  tube  is  equal 


534  LOSSES  AND  MECHANICAL   EFFECT. 

to  v2',  the  velocity  with  which  the  water  leaves  the  turbine. 
Also  let  A»  be  the  sectional  area  of  the  tube.      Then 

o 


f&(=f^  being  the  coefficient  of  friction  with  an  average  value 
of  .0067,  /'  the  length  of  the  tube,  and  m  its  mean  hydraulic 
depth. 

VII.    The   loss   (Z-7)  of  head  due  to  entrance  to  sluice   at 
base  of  tube.      This  loss  may  be  expressed  in  the  form 

7         ,EL 

~~ 


the  average  value  of/7  being  about  .03. 

VIII.    The  loss  (Z-8)  of  head  due  to  the  energy  carried  away 
by  the  water  on  leaving  the  suction-tube. 


and  v^  usually  varies  from  \  V^gH  to  f  \/2gH. 

In  good  turbines  the  loss  should  not  exceed  6  per  cent. 
It  might  be  reduced  to  3  per  cent,  or  even  to  I  per  cent,  but 
this  would  largely  increase  the  skin-friction. 

IX.  The  loss  of  head  (X9)  produced  by  the  friction  of  the 
bearings. 


fj.  being  the  coefficient  of  journal  friction,  Wthe  weight  of  the 
turbine  and  of  the  water  it  contains,  and  p  the  radius  of  the 
journal. 

Hence  the  total  loss  of  head 

=  L1  +  Li  +  L3  +  Lt  +  Ll  +  Lt  +  Lr  +  Ls  +  L,  =  L, 
and  the  total  mechanical  effect 


EXAMPLE.  535 

NOTE. — If  there   is    no    suction-tube,    L5  =  o  =  L6  =  L^ 
=  Zg,  and  the  total  loss  becomes 

r     ,    r         T         r         T     \v?        j  fal1    from    outlet-surface    to 

^i  -f  ^2  -f  ^3  4  ^4   ,   A,  +  ^  H-  {      tail-water  surface. 

EXAMPLE. — A  vortex  turbine,  with  a  draft-tube  of  the 
same  sectional  area  as  that  of  the  outlet-orifice  openings,  passes 
100  cu.  ft.  of  water  per  second  under  the  head  of  9^  ft.  The 
exterior  and  interior  diameters  are  in  the  ratio  of  5  to  4,  and 
the  outlet-  and  inlet-areas  are  in  the  ratio  of  9  to  10.  The 
direction  of  the  water  at  the  inlet  and  the  outlet  lip  angle  are 
given  by  sin  y  =  .25  =  sin  ft.  The  water  leaves  the  tube 
through  a  sluice  having  a  sectional  area  10  per  cent  greater 
than  that  of  the  outlet-orifice  area.  The  outlet  velocity  of  whirl 
is  nil,  i.e.,  d  =90°. 

Disregard  the  losses  Ll ,  Z3 ,  L5 ,  L6 ,  and  Lg. 

v  2 
The  loss  of  head  to  inlet  =f2-±-. 

V* 
44       "    "      "     in  wheel-passages          =  f^—2-. 

o 
V  2 

"       "    "      "     at  sluice-entrance  =/-*-. 


"       "    "      "     carried  away  by  water  =       — . 
Hence  the  total  loss  of  head 


But,  by  the  condition  of  continuity, 

^^  sin  r  =  g  =  ^2F2  sin  ft  =  Aft. 

Therefore 

Vl    _A2    sin^_  ^4_^_2sil._^5 

T^^T/STJ;-       v~  A*       1.1 


S36  EXAMPLE, 

Hence  the  total  loss  of  head 


taking  /2  =  .1,  /4  =  .126,  and  /7  =  .03. 
Again,  the  "useful"  head 

=    — —  =  -  -  .—  .#,.#,  cos  v 
32          32     4 

5  ^42F2     sin/?  cos  v       F2     135 

=    -~  PI  COS  /?  .    -V^  .  ~  — —   =  — -    .   -/^-  =  2.109375   ft. 

128    2  Ai  smy  2g      64 

Therefore 

9.5  =  —(.26  +  2.1094)  =  ~ ~  X  2.3694, 

y   j  s*  ,_\  -7~/          /-» n-  **    s^' 


or 


F2 

—  ^  =  4  ft.,  approx.,     and     F2  =  16  ft.  per  sec. 

The  useful  work  per  Ib.  of  water 

=  4X^  =  8.4375  ft.-lbs. 
The  work  consumed  in  hydraulic  resistances  per  Ib. 

=  4  X  .26  =  1.04  ft.-lbs. 
The  total  work  per  Ib.  of  water  =  9-4775- 
The 


=  ^?Cosec/S=^  =  25  sq.  ft.  and  ^,  =       =  27.78  sq.  ft. 


EXAMPLE. 
Again, 

v\  ~  -9^2  =  J4-4  ft.  per  sec., 
«2  =  F2cos/?  =  l6V^=  1  5-492  ft.  per  sec., 
and 


ft-  per  sec. 


Also, 

ul       sin  (<*  -|-  y) 


or 


Ul 

~ 


=  ^  (  -f~  X4-  3-8/3=  1.5061. 

and  «=33035'- 

If  the  diameter  of  the  tube  is  equal  to  that  of  the  outlet- 
surface,  viz.,  4  ft.,  and  if  its  lower  edge  is  rounded  so  that 
/7  =  O,  then 


energy  per  Ib.  of  water  carried  awav  =  ~^- 


_   V} 


.  015448. 


The  loss  in  shock  in  draft-tube 

F,'  i   /        I75^2      F/ 
:  ^?6  I1  "^  ==  ^  X  - 

Thus  the  total  loss  now  becomes 

F2  772 

2^-081  +  .126  +  .05448  +  .04705)  =  ~X  .2695. 

V* 

As  before,  the  useful  head  =  —  2-  x  2.  1094. 

o 


538  EXAMPLE. 

V* 
Therefore  the  total  head    =  ~^-  X  2.3789, 

2. 1094 
and  the  efficiency  =  2       g    = 

Also, 

o  c  =  -2-X2.378o,  or  -*-=  3.993  ft.,  and  V2=  15. 987ft.  persec. 
2£-  2£- 

v  2 
If  there    is    no    draft-tube,     —     must    be    substituted    for 

-  *  vj_  __  V?  JL 

Thus  the  total  loss  of  head  is  now 


L2.(.o8i  +  .126  +  .0625)  =  -^  X  -2695, 

F2 
which  exceeds  the  loss  of  head  with  a  draft-tube  by  —  X  .0095 

=  .038  ft.,  which  is   less   than  four  hundredths  of  a  foot  and 
is  practically  inappreciable. 


EXAMPLES.  539 


EXAMPLES. 

1.  A  downward-flow  turbine  of  24  ins.  internal  diameter  passes  10 
cu.  ft.  of  water  per  second  under  a  head  of  31  ft.;  the  depth  of  the  wheel 
is  i  ft.  and  its  width  6  ins.     Find  the  efficiency,  assuming  the  whirling 
velocity  at  outlet  to  be  nil.  Ans.  .997. 

2.  A  downward-flow  turbine  of  5  ft.  external  diameter  passes  20  cu. 
ft.  of  water  per  second  under  a  head  of  4  ft.,  the  depth  of  the  wheel 
being  i  ft.     The  water  enters  the  wheel  at  an  angle  at  60°  with  the  ver- 
tical, the  receiving-lip  of  the  wheel-vanes  is  vertical,  and  the  velocity  of 
whirl  at  outlet  is  nil.     Find  the  internal  diameter  and  the  speed  in  rev- 
olutions per  minute.  Ans.  4.6  ft.;  46.53. 

3.  A  downward-flow  turbine  has  an  internal  diameter  of  24  ins.  ;  the 
breadth  of  the  wheel  is  6  ins. ;  the  turbine  passes  33  cu.  ft.  per  second 
under  an  effective  head  of  16  ft.     Assuming  the  whirling  velocity  at  out- 
let to  be  nil,  find  the  efficiency  and  power  of  the  turbine.     If  the  vane- 
lip  at  inlet  is  vertical,  find  the  direction  of  the  vane  at  outlet,  and  the 
speed  of  the  turbine  in  revolutions  per  minute. 

Ans.  .931  ;  55.865  H.P.  ;  0  =  y  =21°  2'^  166.7. 

4.  Discuss  the  preceding  example  on  the  assumption  that   the  pe- 
ripheral speed  at  outlet  (wa)  is  equal  to  the  speed  of  the  water  at  that 
point  relatively  to  the  wheel  ( F2). 

Ans.  .928  ;  55.715  H.P. ;  ft  =  21°  47'  and  y  =  20°  21'. 

5.  An  axial-flow  impulse  turbine  of  5  ft.  mean  diameter  passes  170 
cu.  ft.  of  water  per  second  under  an  effective  head  of  8.6  ft. ;  the  depth 
of  the  wheel  is  .9  ft.    At  what  angle  should  the  water  enter  the  wheel  to 
give  an  efficiency  of  81  per  cent,  the  width  of  the  wheel  being  constant 
and  disregarding  hydraulic  resistances?     z/w"=o.  Ans.  =  27°  16'. 

Also  find  («)  the  velocity  with  which  the  water  enters  the  wheel; 
(ft)  the  speed  of  the  turbine  in  revolutions  per  minute;  (c)  the  directions 
of  the  vane-edges  at  inlet  and  outlet ;  (d)  the  velocity  of  the  water  as  it 
leaves  the  wheel ;  (e)  the  power  of  the  turbine. 

Ans.   (a)     23.46    ft.    per   second ;    (6)   45.08 ;    (c)   a  —  130°    10'; 
ft  =  42°  19' ;  (d)  10.748  ft.  per  second  ;  (e)  148.65  H.P. 
If,  instead  of  assuming  that  the  whirling  velocity  at  exit  is  nil,  it  is 
assumed  that  the  peripheral  speed  («,)  of  the  wheel  at  the  mean  radius 
is  equal  to  the  relative  velocity  ( Fa)  of  the  water  at  exit,  show  how  the 
several  results  are  affected. 

Ans.    Y  =  25°   8'  ;    (a)    23.46   ft.    per   second  :    (b)    54.638 ;    (c) 
a  =  124°  49',  ft  =  44°  6';  (d)  10.748  ft.  per  second  ;  (e)  148.65  H.P. 


540  EXAMPLES. 

Also  show  how  the  results  are  affected  when  it  is  assumed  that  the 
hydraulic  resistances  necessitate  an  increase  of  12^  per  cent  in  the  head 
equivalent  to  the  velocity  with  which  the  water  enters  the  wheel,  and  an 
increase  of  10  per  cent  in  the  head  equivalent  to  the  relative  velocity 
( F8)  at  outlet. 

Ans.   When  i /w"  =  o  (a)  22.12  ft.    per  second;   (£)   44.21;   (c) 
a  =  147°  50',  /3  =  27°  44;    (d)  10.748  ft.  per  second  ;  (e)  148.65  H.P. 
When  Ui  =  V*  (a)  22.119  ft«  Per  second  ;  (b)  50.97;  (c)  a=  123°  19', 
ft  =  47°  28' ;  (d)  10.748  ft.  per  second  ;  (*)  148.65  H.P. 
If  the  turbine  has  65  guide-blades  of  .2-in.  thickness  and  63  wheel- 
vanes  of  .4~in.  thickness,  find  the  widths  of  the  inlet  and  outlet  openings. 
Ans.  If  vw"  =    o,    d\  —  4.214  ft.,  dt  =  2.83  ft. 
If  u,    =  F3 ,  d,  =  1.78  ft.,     d,  =  1.48  ft. 

6.  The  efficiency  of  an  axial-flow  turbine  of  4  ft.  mean  diameter  is 
90  per  cent,  and  it  passes  12  cu.  ft.  per  second  under  an   effective  head 
of  40  ft.     At  the  mean  radius  the  water  enters  at  an  angle  of  30°  with 
the  wheel's  face,  and  the  whirling  velocity  at  outlet  is  nil.     Find  (a)  the 
velocity  with  which  the    water   enters   and    leaves  the  wheel ;  (£)  the 
directions  of  the  vane  .at  inlet  and  outlet  ;  (c)  the  sectional  areas  of  the 
inlet-  and  outlet-orifices ;  (d)  the  speed  of  the  wheel  in  revolutions  per 
minute  ;  (<?)  the  power  of  the  turbine. 

Ans.  (a)  32  ft.  per  second,  16  ft.  per  second  ;    (b)  a  =  49°  6', 
ft=  21°  3';  (c)  .75  sq.  ft.;  (d)  198.39;  (*)  49iT  H.P. 

7.  An  axial-flow  turbine  of  5  ft.  mean  radius  passes  212  cu.  ft.  of 
water  per  second  under  a  total  effective  head  of  12.1  ft.     At  the  mean 
radius,  the  direction  of  the  inflowing  water  makes  an  angle  of  70°  with 
the    vertical,   and  the   vane-lip  at   the   outlet  makes  an  angle   of    17° 
with  the  wheel's  periphery.     If  the  whirling  velocity  at  the  outlet-sur- 
face is  nil,  find  (a)  the  velocity  with  which  the  water  must  enter  the 
wheel  to  give  an  efficiency  of  .953  per  cent.      Also  find  (b)  the  direction 
of  the  vane-lip  at  outlet ;  (c}  the  speed  ^ot  the  wheel  in  revolutions  per 
minute;    (d)  the  widths  and  areas  of  the  inlet- and  outlet-orifices;  (<?) 
the  power  of  the  turbine. 

Ans.  (a)  19.9  ft.  per  second  ;  (b)  a  —  81°  25';  (c)  37.67  ;  (d)  .991   ft., 
31.148  sq.  ft.,  1.81  ft.,  35.14  sq.  ft.;  (<?)  277.709. 

If  the  turbine  has  41  guide-blades  and  40  wheel-vanes,  all  of  .25  in. 
thickness,  find  the  widths  of  the  inlet-  and  outlet-openings. 

Ans.   1.23  ft.;  1.37  ft. 

8.  Write  down  the  equations  for  Jouval's  modification  of  Euler's  turbine. 

9.  An  axial-flow  impulse  turbine  passes  170  cu.  ft.  of  water  per  second 
under  an  effective  head  of  9.5  ft.,  the  depth  of  the  wheel  being  .9  ft.  and 
its  mean  radius  4.2  ft.     The  vane-lip  at  the  outlet  makes  an  angle  of  72° 
with  the  vertical.     Assuming  that  the  whole  of  the  effective  head    is 
transformed  into  useful  work,  and  that  the  whirling  velocity  at  the  outlet- 
surface  is  nil,  find  (a]  the  inclination  to  the  horizontal  of  the  outlet-lip  of 


EXAMPLES.  541 

the  guide-vane  :  (b)  the  direction  of  the  inlet-lip  of  the  wheel-vane  ;  (c]  the 
efficiency  ;  first  neglecting  hydraulic  resistances,  and  second  taking  these 
resistances  into  account. 

Ans.    First.       (a)  59°  52';   (£)  60°  16'  ;  (c)  .905. 
Second,  (a)  47°  21'  ;  (£)  85°  18';  (c)  .804. 

10.  In  the  preceding  example  find  the  inlet-  and  outlet-orifice  areas 
in  the  two  cases. 

Ans.    First.  8.38  sq.  ft.  ;  22.4    sq.  ft. 

Second.    10.45  sq-  '&•  J  32-°8  sq.  ft. 

If  there  are  62  wheels  and  66  guide-vanes,  the  thickness  of  the  latter 
being  .2  in.  and  of  the  former  .4  in.,  find  the  width  of  the  inlet-orifices. 
Ans.   First.   .409  ft.;  1.26  ft.     Second.   .508  ft.;  1.81  ft. 

11.  An  axial-flow  turbine  passes  200  cu.  ft.  of  water  per  second  under 
a  head  of  14  ft.,  the  depth  of  the  wheel  being  i  ft.     The  mean  radius  of 
the  wheel  is  3  ft.;  the  areas  of  the  inlet-  and  outlet-surfaces  are  in  the 
ratio   of  7  to  8  ;   the  water  enters  the  wheel  at  an  angle  of  21°  to  the 
wheel  face,  and  the  outlet  edge  of  the  vane  makes  an  angle  of  16°  with 
the  face.     Find  the  speed,  efficiency,  and  power  of  the  turbine,  and  also 
the  direction  of  the  inlet-lip  of  the  vanes,  v^'  =  o. 

Ans.   73.69    revolutions    per    minute;     .954;    325.243    H.P.  ; 
a  =  65°  57'. 

12.  A  downward-flow  turbine  of  3T9T  ft.  mean  diameter   and  of  the 
impulse  type  is  supplied  with  5^  cu.  ft.  of  water  per  second  under  a  head 
of  400  ft.  and  makes  500  revolutions  per  minute.     The  water  enters  the 
wheel  at  an  angle  of  sin-1  .6  with  the  horizontal,  and  the  depth  of  the 
wheel  is  i  ft.     The  water  leaves  the  turbine  with  a  velocity  of  60  ft.  per 
second.     Determine   the  whirling  velocity  at  outlet,  the    direction    in 
which  the  water  leaves  the  turbine,  the  efficiency,  and  the  horse-power. 

Ans.  17.725  f/s;  72°  49';  .86;  214.8. 

13.  In  an  A.  F.  impulse  turbine  of  4  ft.  diameter,  i  ft.  deep,  and  with 
a  6-in.  width    of   opening   at   inlet   and   outlet,  the  efficiency  (77)  —  .8  ; 
ft  —  30°  ;  y  =  30°  ;  V*  =  u*.    Determine  the  inlet-lip  angle  (a),  the  effec- 
tive fall,  the  delivery  (0  (disregarding  thickness  of  vanes),  the  H.P.  and 
the  number  of  revolutions  per  minute. 

Ans.  a  =  75°;  1.366  ft.;  29.39  cu.  ft-  Per  second;  6.322  H.P.  ; 


14.  An  axial-flow  reaction  turbine  of  7  ft.  mean  diameter  passes  198 
cu.  ft.  of  water  per  second  under  a  total  head  of  13.5  ft.,  the  depth  of  the 
wheel  being  i  ft.     At  the   inlet-surface  the  vane-lip  is  vertical  and  the 
water  leaves  the  wheel  vertically.     If  the  inlet  width  of  the  wheel  is  i  ft. 
and  the  outlet  width  i£  ft.,  find  the  direction  in  which  the  water  enters 
the  wheel,  the  direction  of  the  lip  at  outlet,  the  inlet  and  outlet  areas, 
the  H.P.  of  the  turbine,  and  its  efficiency. 

Ans.  24°  4'  ;  19°  40'  :  22  and  27$  sq.  ft.;  285.525  ;  .94. 

15.  An  axiai-flow  turbine  is  to  be  used  tor  raising  water.     Explain 


542  EXAMPLES. 

how  the  vanes  should  be  arranged,  and  show  how  to  determine  the 
efficiency. 

16.  In  an  A.  F.  impulse  turbine,  working  under  a  head  of  100  ft.,  the 
direction  in  which  the  water  enters  at  the  mean  radius  makes  an  angle 
of  23°  16'  with  the  vertical  and  leaves  the  wheel  without  velocity  of 
whirl.     The  depth  of  the  wheel  is  i   foot,  and  the  inlet  velocity  (z/i)  is 
equal  to  the  linear  velocity   (HI)  of   the  wheel's  surface  at    the    mean 
radius.     The  mean  diameter  of  the  wheel  is  3^  ft.,  and  its  width  is  6  ins. 
Find  the  blade  angles  at  inlet  and  outlet,  the  efficiency,  the  speed  in 
revolutions  per  minute,  the  amount  of  water  passing  through  the  turbine 
per  second,  and  the  H.P. 

Ans.  56°  38';  64°  52';  .782;  436^,  404^  cu.  ft.;  3592f. 

17.  Water  is  delivered  to  an  O.  F.  turbine  at  a  radius  of  24  in.  with  a 
whirling  velocity  of  20  ft.  per  second,  and  leaves  in  a  reverse  direction  at 
a  radius  of  4  ft.  with  a  whirling  velocity  of  10  ft.  per  second.    If  the  linear 
velocity  of  the  inlet-surface  is  20  ft.  per  second,  find  the  head  equivalent 
to  the  work  done  in  driving  the  wheel.  Ans.  24.8  ft. 

18.  An  outward-flow  turbine  of  9.5  in.  internal  diameter  works  under 
an  effective  head  of  270  ft.     Find  the  speed  in  revolutions  per  minute, 
assuming  that  the  whirling  velocity  at  the  inlet-surface  relatively  to  the 
wheel  is  nil  and  that  the  efficiency  is  unity.  Ans.  2242. 

19.  An  outward-flow  turbine,  whose  external  and  internal  diameters 
are  8  ft.  and  5^  ft.  respectively,  makes  26  revolutions  per  minute  urrder 
an  effective  head  of  4  ft.      The  water  enters  the  wheel  in  a  direction 
making  an  angle  of  30°  (y)  with  the  direction  of  motion  at  the  point  of 
entrance.      Determine    the  angles  of  the    moving  vane  at    ingress   and 
egress,  the  efficiency  being  .85.     Also  find  the  energy  per  pound  of 
water  carried' away  by  the  water  as  it  leaves  the  turbine,  vw"  =  o. 

Ans.   a  =  130°  2' ;  ft  =  29°  38' ;   .6  ft.-lbs. 

20.  A  radial  outward-flow  turbine  of  the  impulse  type  passes  8^  cu.  ft. 
of  water  per  second  under  an  effective  head  of  560  ft.;  the  width  of  the 
wheel  is  7^  in.  ;  the  radius  to  the  outlet-surface  is  1.15  times  the  radius 
to  the  inlet-surface  ;  the  linear  velocity  of  the  inlet-surface  is  87  ft.  per 
second  ;  the  direction  of  the  water  at  entrance  makes  an  angle  of  17° 
with  the  wheel's  periphery.     Find  (a)  the  efficiency  ;  (b)  the  lip  angles  ; 
(c)  the  areas  of  the  inlet-  and  outlet-orifices,  neglecting  first  hydraulic 
resistances,  and  second  taking  these  resistances  into  account  (i/w"  —  o). 

Ans.  First,  (a)  .879;  (b)  a  =149°  31'  and  /?  =  33°  21',  (c) 
•1S3S  scl-  ftv  and  .1291  sq.  ft.  Second,  (a)  .767;  (<£)  a  =  153°  44' 
and  ft  =  28°  55';  (c)  .  1 76  sq.  ft.  and  .154  sq.  ft. 

21.  Construct  a  Fourneyron  turbine  for  a  fall  of  5  ft.  with  30  cu.  ft.  of 

water  per  second,  a  =  80°,^  =  30°,  —  =  1.35.     Assume   «2  =  F2,  and 

neglect  hydraulic  resistances. 

Ans.  ft  =  16°  42'  ;  Ai  =  4.29  sq.  ft.:  Ai  =  5.8189  ft.;  17  =  .915  ; 
if  r^  —  1.8  ft.,  then  di  =  d*  ='.38  ft. 


EXAMPLES.  543 

22.  In  an  impulse  outward-flow  turbine  of  10  B.H.P.  ,working  under 
a  head  of  9  ft.,  y  =  22^° ;  180°  —  a  —  37^°  ;  (3  =  45° ;  9(ra  -  ri)  =  n ; 
di  =  .2ri.  There  is  a  loss  of  5  per  cent  due  to  friction  in  the  velocity  at 
entrance.  Find  the  efficiency  (rf),  the  volume  of  water  passed  per 
second,  and  the  diameter  of  the  turbine. 

Ans.  .705;  I3.869CU.  ft.;  2.249  ft. 

25.  A  turbine  delivers  i  cu.  ft.  of  water  per  second.  The  water 
leaves  the  outlet  periphery  radially  (vw"  =  o).  The  vane-lip  at  inlet  is 
radial  (cc  =  90°).  The  direction  of  inflow  makes  an  angle  of  60°  with  the 
wheel's  periphery.  The  radius  of  inlet-surface  is  2  ft.  The  number  of 
revolutions  per  minute  is  roo.  If  the  efficiency  is  90  per  cent,  find  the 
head  and  the  effective  work  done.  Ans.  15.243  ft. ;  1.5625  H.P. 

24.  One  cubic  foot  of  water  per  second  enters  a  radial  O.F.  impulse 
wheel  of  2  ft.  external  and  i|  ft.  internal   diameter,  at  an  angle   of  60° 
with  the  radius,  and  leaves  without  whirl.     The  effective  head  is  400  ft. 
The  peripheral  speed  at  the  outlet-surface  is  20  ^3  ft.  per  second.     De- 
termine   a, 'Vi ,  the  outlet  and   inlet  areas   and    depths,    the  H.P.   and 
efficiency.        Ans.   1.15  sq.  ins.,  1.8  sq.  ins.;  183  ins.,  .39  ins.;  12.8;  .28. 

25.  In  a  radial-flow  reaction  turbine  with  radial  inlet-lips,  if  di  =  2^ 
and  y  =  tan~  l  4,  show  that  the  reciprocal  of  the  efficiency  is  i  +  tan  ft 
if  the  whirling  velocity  at  outlet  is  nil. 

26.  An  O.F.  impulse  turbine  of  3^  ft.  exterior  and  3  ft.  interior  diam- 
eter passes  100  cu.  ft.  of  water  per  second  under  a  head  of  625  ft.     At 
entrance  the  direction  of  the  water  makes  an    angle  of  30°  with  the 
periphery.     If   the  relative  and    peripheral    speeds  at  outlet  are  equal, 
determine  the  direction  and  magnitude  of  the  velocity  of  the  water  on 
leaving  the  wheel,  the  efficiency,  and  the  speed  in  revolutions  per  min- 
ute.    Disregard  hydraulic  resistances. 

Ans.  If   di  =  d* ,  v*  =  91.065  ft.  per  sec. ;  8  =  70°  14^' ;  rj  =  .79;  N=  734.8. 
If  Ai  =  A*,Vv=:  109.435  "        "        ;  d  —  66°  02' ;  rj  —  .70 ;  TV  =  734.8. 

27.  A  radial  impulse   turbine  passes   8|  cu.  ft.  of  water   under   an 
effective  head  of  560  ft.     The  direction  of  the  entering  water  is  inclined 
at  17°  to  the  wheel's  periphery.     The  linear  speed  of  the  inlet-surface  is 
87  ft.  per  second.     Assuming  that  the  velocity  of  whirl  at  the  outlet  is 
nil,  and  disregarding  hydraulic  resistances,  find  (a)  the  efficiency;  (&)  the 
velocity  with   which  the  water  enters  the  wheel ;  (c)   the   velocity   of 
the  water  as  it  leaves  the  wheel ;  (d)  the  sectional  areas  of  the  inflow- 
ing and  outflowing  stream;    (e)  the  direction  of  the  vane-lip  at  inlet; 
(/")  the  power  of  the  turbine. 

The  radii  of  the  inlet-  and  outlet-surfaces  are  4^  ft.  and  4^  ft.  re- 
spectively. Find  (£•)  the  direction  of  the  vane  edge  at  outlet. 

Ans.  (a}  .879;  (b)  189.31  ft.  per  sec. ;  (c]  65.86  ft.  per  sec.  ; 
(d)  .15356  sq.  ft.,  .129  sq.  ft.;  (<?)  a—  149°  31'; 
(/)  47543  H.P.;  (jg)  ft  =  41*  $'• 

28.  In  the  preceding  example  show  how  the    results  are  affected  by 


544  EXAMPLES. 

taking  .94  as  the  coefficient  of  velocity  in  calcui  *ting  the  velocity  with 
which    the  water  enters   the   wheel,  and   assuming  that  —   — -    is   the 

10     2g 

frictional  loss  of  head  in  the  passages. 

Ans.  (a)  .826;  (b)  177-955  ft-  per  sec.;  (c)  36.7348  ft.  per  sec.; 
(d)  .163  sq.  ft.;  .23i4sq.  ft.  ;  (e)  a  =  147°  57'; 
(/)  446.9  H.P.  \(g)fl=  25°  54'. 

29.  In  an  I.F.  turbine  the  radius  of  the  inlet-surface  is  twice  that  of 
the  outlet-surface ;  the  linear  velocity   of  the  inlet-surface   is  one  half 
that  due  to  the  head ;  the  water  enters  the  wheel  with  a  velocity  of  flow 
(Vr)  equal  to  one  eighth  that  due  to  the  head,  and  the  sectional  area  of 
the  water-way  is  constant  from  inlet  to  outlet.     Find  the  angle  between 
the  discharging  lip  of  the  vane  and  the  wheel's  periphery,  the  whirling 
velocity  at  the  outlet-surface  being  nil.  Ans.  Cot~  J  2. 

30.  In  a  vortex  turbine  the  depth  of  the  inlet-orifices  is  one  eighth 

of  the  diameter  of  the  wheel  f  =  — 1j  and  —  of  the  depth  of  the  outlet- 
orifices.     The  width  of  the  wheel  is  one  tenth  of  the  diameter  [  =  — - ). 


The  inlet-lip  of  the  vanes  is  radial,  and  the  water  enters  at  an  angle  of 
30°  with  the  inlet  periphery.  Find  the  size,  speed,  and  efficiency  of  the 
turbine  in  terms  of  the  supply  of  water  Q  and  the  effective  head  H~ 
Also  find  the  direction  of  the  outlet  edge  of  the  vanes. 

Ans.   I.    Assume  vw"  =  o.      Then  r\  =  .458—.; 

H* 

No.  of  revolutions  per  minute  =  109.5 — ; 

77  =  .863;   ft  =  35°  ii'. 
II.   Assume  u*  —  V*.     Then  r\  =  .44—; 

No.  of  revolutions  per   minute  =  122.39 — J 

Qk 

77  =  .8146;   ft  =  44°  48'. 

31.  A  vortex  turbine,  with  a  wheel  of  2  ft.  diameter  and  6  ins.  breadth, 
passes  10  cu.  ft.  of  water  per  second  under  a  head  of  32  feet.     Find  the 
inclination    of   the    guides   and    the   power   of   the    turbine.      Assume 
w2  —  F2 ,  a  =  90°,  and  the  efficiency  =  i.  Ans.  5°  41' ;  36T\  H.P. 

32.  An  inward-flow  turbine  has  an  internal  radius  of  12  ins.  and   an 
external  radius  of  24  ins.  ;  the  water  enters  at  15°  with  the  tangent  to 
the  circumference,  and  is  discharged  radially ;  the  velocity  of  radial  flow 
is  5  ft.  at  both  circumferences;  the  velocity  of  outer  periphery  of  wheel 
is  16  ft.   per  second.     Find  the  angles  of   the   vanes  at  the  inner  and 
outer  circumferences,  and  the  useful  work  done  per  pound  of  fluid. 

Ans.  0  =  12°;  d=uV  i';  9.35  ft.-lbs. 


EXAMPLES.  545 

33.  For  a  supply  of  64  cu.  ft.  per   second,  under  a   head  of   81    ft., 
determine  the  speed,  size,  H.P.,  and   efficiency  of  a  vortex  turbine  in 
which    d\  =   ri  —  3^/2  =  5  x  width  of  wheel,  assuming  that  there  is  no 
velocity  of  whirl  at  outlet. 

34.  A  radial  I.  F.  reaction  turbine,  with  or  without  draft-pipe,  passes 
113  cu.  ft.  of  water  under  an  effective  head  of  13  ft.     The  radius  of  the 
inlet-surface  is  1.169  times  the  radius  of  the  outlet-surface,  and  the  ratio 
of  the  outlet  to  the  inlet   area  is  .92.      The  vane-lip  at   outlet  makes 
an  angle  of  15°  with  the  wheel's  periphery,  and  the  water  enters  at  an 
angle  of   12°  with   the  wheel's  periphery.     The   sectional  area  of  the 
draft-tube  (if  there  is  one)  at  the  point  of  discharge  is  1.035  times  the  sec- 
tional area  of  the  outlet-orifice.    Show  that  the  useful  work  per  pound  of 
water  is  1 1.117  ft.-lbs.,  and  that  the  work  consumed  in  hydraulic  resistance 
(Art.  8,  page  531)  is  nearly  1.882  ft.-lbs.;  also  find  .,4,,  ^42,  7/2,  and  the 
efficiency. 

Ans.  (a)  28.2975  sq.  ft.;  26.03  sq.  ft.;  (b)  4.34  ft.  per  sec.;  .855. 

35.  In  the  preceding  example,  if  the  radius  of  the  outlet-surface  is  4 
ft.,  find  (a)  the  speed  of  the  wheel  in  revolutions  per  minute  ;  also  find 
(b)  the  depth  of  the  wheel  at  inlet  and  outlet,  the  guide-vanes  being  40 
and  the  wheel-vanes  41  in  number,  and  the  thickness  of  the  former  being 
T3ff  inch  and  of  the  latter  ±  inch.        Ans.  (a}  38.656;  (b)  1.23  ft,,  1.35  ft. 

36.  In  example  34  find   the  efficiency  if  the  diameter  of  the  draft- 
tube  is  made  the  same  as  the  diameter  of  the  outlet-surface,  the  lower 
edge  of  the  tube  being  rounded.     What  will  be  the  "  loss  in  shock  "  in 
the  tube  per  pound  of  water  ?  Ans.  .864  ;  .077  ft.-lbs. 

37.  An  inward-flow  turbine  has  an  external  diameter  of  3  ft.  and  an 
internal  diameter  of  2  ft.     It  passes  12  cu.  ft.  of  water  per  second  under 
an  effective  head  of  40  ft.    The  water  enters  the  wheel  at  an  angle  of  30° 
with  the  wheel's  periphery,  and  the  depth  of  the  outlet-orifices  is  twice 
the  depth  of  the  inlet-orifices.     The  efficiency  of  the  turbine  is  .9.     Dis- 
regarding friction,  find   (a}  the  vane-angles  at  inlet  and  outlet ;  (b]  the 
velocity  with  which  the  water  leaves  the  wheel ;  (c)  the  speed  of  the  tur- 
bine in  revolutions  per  minute  ;  (d)  the  velocity  with   which  the  water 
enters  the  wheel ;  (e)  the  areas  of  the  outlet-  and  inlet-orifices  ;  (/)  the 
power  of  the  turbine  (vw"  =  o). 

Ans.  (a)  a  =  105°  09',  ft  =  35°  35'  ;  (b)  16  ft.  per  sec.;  (c)  198.39 ; 
(d)  42!  ft.  per  sec. ;  (e)  .5625  sq.  ft.  ;  .75  sq.  ft.  ;  (/)  49^  H.P. 

38.  In  an   inward-flow  reaction  turbine  of  6.27  H.P.  the  radial  veloc- 
ity of  flow  is  constant  from  inlet  to  outlet  and  is  12  ft.  per  second.     The 
water,  with  a  velocity  of  60  ft.  per  second,  enters  at  1 1°  32'  with  the  wheel's 
periphery,  which  has  a  linear  speed  of  50  ft.  per  second.     The  diameters 
of  the  outlet- and  inlet-surfaces  are  i  and  2  ft.  respectively.     Find  the  tip 
angles,  the  head,  the  efficiency,  and  the  quantity  of  water  passing  through 
the  turbine  per  second. 

Ans.   a—  126°  12';  ft  —  151°  19';  91.86  ft.;  6o# ;   I  cu.  ft. 


546  EXAMPLES. 

39.  An  inward-flow  radial  impulse  turbine  of  4.5  ft.  and  4  ft.  external 
and  internal  radii  passes  8|  cu.  ft.  of  water  per  second  under  an  effective 
head  of  560  ft.     The  direction  of  the  entering  water  is  inclined  at  17°  to 
the  wheel's  periphery,  and  the  wheel  has  the  same  depth  at  the  inlet- 
and  outlet-surfaces.     If  the  peripheral  speed  at  the  outlet-surface  (;/a) 
is  equal  to  the  relative  velocity  of  the  water  (  F2)  with  respect  to  the 
wheel,  find  (a)  the  efficiency ;  (b)  the  speed  of  the  turbine  in  revolutions 
per  minute;  (c]   the  sectional  areas  of  the  stream  at  inlet  and  outlet; 
(d)  the  direction  of  the  vane-outlet  edge;  (<?)  the  velocity  of  the  water 
as  it  leaves  the  wheel  ;  (/)  the  power  of  the  turbine. 

Ans.  (a)   .873;    (b)  209.94;    (c)    .15357    sq.    ft.,   .  13651-  sq.   ft.; 
(it)  ft  =  45°  2' ;    (e)  67.39  ft.  per  second  ;    (/)  472.33  H.P. 

40.  In    the   preceding   example   examine   how    the    results    will    be 
affected  when  hydraulic  resistances  are  taken  into  account,  allowing  .94 
as  a  coefficient  of  velocity  for  the  water  on  entering  the  wheel,  and  as- 
suming that  the  head  equivalent  to  the  relative  velocity  ( K2)  on  leaving 
the  wheel  is  increased  by  10  per  cent. 

Ans.  (a). 865  ;  (b}  193.185  revolutions  per  minute  ;  (c)  .163  sq.  ft., 
.145  sq.  ft.;  (d)  /3  =  46'  18';  (e)  63.653ft.  per  second  ;  (/)  467.83  H.P. 

41.  An   I.    F.    turbine  of  4  ft.    external    diameter    works    under  an 
effective  head  of  250  ft.     Find  the  speed  of  the  wheel  in  revolutions  per 
minute,  vw"  being  o,  the  efficiency  unity,  and  a  =  90°.  Ans.  427. 

42.  An    I.  F.  turbine   of  4   ft.  external   and   3  ft.    internal    diameter 
makes  360  revolutions  per  minute.    The  sectional  area  of  flow  is  3  sq.  ft. 
and  is  the  same  in  every  part  of  the  turbine.     The  direction  of  the  in- 
flowing water  makes  an  angle  of  30°  with  the  wheel's  periphery.   Assum- 
ing that   the   whirling  velocity  at  the  outlet-surface  is  nil,  find  (a)  the 
efficiency;  (b)  the  H.P.  ;  and  (c)  the  delivery  in  cubic  feet  per  minute. 
The  total  head  is  200  ft.  Ans.  (a)  .86  ;  (b)  2476.8  ;  (c)  7593. 

43.  An  inward-flow  turbine  being  required  for  an  available  head  of 
20  ft.  and  a  discharge  of  800  cu.  ft.  per  minute,  determine  (a)  the  size 
and  (b)  the  speed  of  the  wheel  ;  (c)  the  inclinations  of  the  guide-  and 

Wheel-vanes  ;  and  (d)  the  efficiency  of  the  turbine,  assuming  r<2  =  |r,  — 
depth  of  wheel  ;  ?V  =  \V?gH \  v™"  =  o,  a  =  90°,  and  d\  =  d*. 

Ans.  (d)  ri  =  .487  ft.,  r\  —  .974  ft.;  (b)  240  revolutions  per  min- 
ute ;  (c)  y  =  10°  21',  ft  —  36°  8';  (d)  93$  per  cent. 

44.  A  vortex  turbine  passes  Q  cu.  ft.  of  water  per  second  under  an 
effective  head  of  H  ft.    The  inlet-lip  of  the  vanes  is  radial,  and  the  direc- 
tion of  the  entering  water  makes  an  angle  of  20°  17'  with  the  wheel's. 

periphery.   The  areas  of  the  inlet- and  outlet-orifices  are         J    and          ' 

respectively,  and  the  width  of  the  wheel  is  — ,  ZV being  the  diameter  of 

the  inlet-surface.     If  the  whirling  velocity  at  the  outlet-surface   is  nil, 
find  («)  the  efficiency  ;  (b)  the  direction  of  the  outlet  edge  of  the  vane ; 


EXAMPLES.  547 

(c)  the  velocity  with  which  the  water  enters  and  leaves  the  wheel ;  (df  >•• 
the  speed  of  the  wheel  in  revolutions  per  minute;  (<?)  the  diameters  o£ 
the  inlet-  and  outlet-surfaces. 

Ans.  (a}  .863;  (6)  ft  =  35°  10';  (c)  6.0677^,   2.9627^; 

H*  O*  Q* 

(</)  109.52-;   (,)  .915^,    .733%- 

45.  A  vortex  turbine  passes   n   cu.  ft.  of  water  per  second  under  a 
head  of  35  ft.  ;  the  diameter  of  the  outlet-surface  is  2  ft.  and  its  breadth 
6  ins.    Find  the  power  of  the  turbine,  disregarding  friction  and  assuming 
that  the  whirling  velocity  at  the  outlet-surface  is  nil. 

Ans.  43-5  H.P. 

46.  Find  the  H.P.  developed   by  an   I.  F.  turbine,  of  3   ft,    external 
and  ig-  ft.  internal    diameter,  passing  900  tons  of  water  per  hour.     The 
velocity  of  whirl  at  inlet  (vwf)  is  equal  to  that  of  the  periphery  and  is  45. 
ft.  per  second ;  the  outlet  velocity  of  whirl  is  i6f  ft.  per  second. 

Ans.  46^. 

47.  A  turbine  with  radial  vanes  passes  3600  gallons  per  hour  under 
an  effective  head  of  36  ft.     Find  the  peripheral  speed  and  the  inlet  area 
so  that  the  efficiency  may  be  a  maximum. 


CHAPTER    VIII. 
CENTRIFUGAL    PUMPS. 

I.  General  Statement. — If  an  hydraulic  motor  is  driven  in 
the  reverse  direction,  and  supplied  with  water  at  the  point  from 
which  the  water  originally  proceeded,  the  motor  becomes  a 
pump.  All  turbines  are  reversible,  and  may  therefore  be 
converted  into  pumps,  but  no  pump  has  yet  been  constructed 
of  an  inward-flow  type.  The  ordinary  centrifugal  pump  is  an 
outward-flow  machine. 

Before  the  pump  can  be  put  into  action  it  must  be  filled, 
and  this  can  be  effected  through  an  opening  (closed  by  a  plug) 
in  the  casing  when  the  pump  is  under  water,  or,  if  the  pump 
is  above  water,  by  creating  a  vacuum  in  the  pump-case  by 
means  of  an  air-pump  or  a  steam-jet  pump,  when  the  water 
must  necessarily  rise  in  the  suction-tube. 

At  first  the  water  rotates  as  a  solid  mass,  and  delivery 
commences  when  the  speed  is  such  that  the  head  due  to  cen- 


trifugal force   ( j   exceeds  the  lift.      This  speed  may  be 

afterwards  reduced,  providing  a  portion  of  the  energy  is  utilized 
at  exit. 

As  soon  as  the  pump,  which  is  keyed  on  to  a  shaft  driven 
by  a  belt  or  other  means,  commences  to  work,  the  water  rises 
in  the  suction-tube  and  enters  the  eye  of  the  pump-disc  on  one 
side,  or  divides  and  enters  on  both  sides. 

As  in  turbines,  the  wheel-blade  tips  are  so  curved  as  to 
receive,  at  a  specified  normal  speed,  the  inflowing  water  with- 
out shock.  The  water  leaves  the  disc  with  a  more  or  less 

548 


CENTRIFUGAL   PUMPS. 


549 


considerable  velocity,  and  impinges  upon  the  fluid  mass  flowing 
around  the  volute,  or  spiral  casing  surrounding  the  disc,  towards 
the  discharge-pipe.  This  volute  should  have  a  section 


FIG.  311. 


FIG.  313. 


gradually  increasing  to  the  point  of  discharge,  in  order  that 
the  delivery  across  any  transverse  section  of  the  volute  may  be 
uniform.  This  volute  is  also  so  designed  as  to  compel  rotation 
in  one  direction  only,  with  a  velocity  corresponding  to  the 
velocity  of  whirl  (vw")  on  leaving  the  fan.  There  are  examples 


35°  CENTRIFUGAL   PUMPS. 

of  pumps  in  which  the  delivery  is  effected  in  all  directions,  and 
the  water  is  guided  to  the  outlet  by  a  number  of  spiral  blades. 

A  centrifugal  pump  is  more  economical  and  less  costly 
for  short  lifts  than  a  reciprocating  pump,  and  has  been  known 
to  give  good  and  economic  results  for  lifts  as  great  as  ^O  ft. 

With  compound  centrifugal  pumps  very  much  greater  lifts 
are  economically  possible. 

There  are  three  main  differences  between  centrifugal  pumps 
and  turbines : 

1st.  The  gross  lift  with  a  pump  is  greater,  on  account  of 
frictional  resistances,  than  the  fall  in  the  case  of  a  turbine. 

2d.  The  water  enters  the  pump-fan  chamber  without  any 
velocity  of  whirl  (rj  =  o),  and  leaves  the  fan  with  a  velocity  of 
whirl  (f,,/')  which  should  be  reduced  to  a  minimum  in  the  act  of 
lifting,  but  which  is  by  no  means  small.  In  a  turbine,  on  the 
other  hand,  the  water  has  a  considerable  velocity  of  whirl  (vj) 
at  entrance,  while  at  exit  the  velocity  of  whirl  (Vw")  is  reduced 
to  a  minimum,  and  is  generally  nil. 

3d.  In  a  turbine  the  direction  of  the  water  as  it  flows  into 
the  wheel  is  controlled  by  guide-blades ;  whereas  in  the  case 
of  a  pump,  the  direction  of  the  water,  as  it  flows  towards  the 
•discharge-pipe,  is  controlled  by  a  single  guide-blade,  which 
forms  the  outer  surface  of  the  volute,  or  chamber,  into  which 
the  water  flows  on  leaving  the  fan. 


FIG.  314. — Experimental  Centrifugal   Pump  in  the  Hydraulic  Laboratory, 

McGill  University. 

Experiment  seems  to  indicate  that  the  efficiency  of  a  centrif- 
ugal pump  increases  as  the  inlet-tip  angle  diminishes,  and  that 


CENTRIFUGAL  PUMPS.  55 r 

it  is  therefore  advantageous  to  make  this  angle  as  small  as  is 
practicable,  but  opinions  on  this  point  differ.  The  real  influ- 
ence of  the  tip  angles  on  the  efficiency  is  yet  to  be  determined, 
•and  it  is  doubtful  whether  the  ordinary  hypothesis  of  radial  flow 
(y  =  90°)  at  inlet  without  shock  is  even  approximately  correct. 

The  inlet  velocity  and  therefore  also  the  pump's  efficiency 
may  be  increased  by  the  use  of  a  suction-tube  with  a  gradually 
diminishing  section,  e.g.,  a  tube  in  the  form  of  the  frustum  of 
a  cone.  A  still  greater  advantage  may  be  obtained  by  giving 
the  discharge-pipe  a  gradually  increasing  section.  In  this  case 
the  velocity  of  discharge  gradually  diminishes  and  the  pressure- 
head  is  proportionately  increased,  so  that  there  is  a  gain  of 
head  available  for  increasing  the  pumping  power.  The 
velocity  in  the  discharge-pipe  should  not  be  too  great,  as  it 
may  lead  to  a  very  sensible  loss  of  energy.  Generally  speak- 
ing, a  velocity  of  3  to  6  ft.  per  second  has  been  found  to  give 
the  most  favorable  results. 

It  is  claimed  by  some  authorities  that  an  advantage  may 
TDC  gained  by  the  addition  of  a  vortex  or  whirlpool  chamber 
surrounding  the  pump-disc.  In  support  of  this  contention  it  is 
urged  that  the  water  discharged  from  the  disc  continues  to 
rotate  in  this  chamber,  and  that  a  portion  of  the  kinetic  energy 
is  thus  converted  into  pressure  energy,  which  would  otherwise 
be  largely  wasted  in  eddies  in  the  volute  or  discharge-pipe 
(see  Art.  21,  Chap.  I).  The  water  leaves  the  vortex-chamber 
with  a  diminished  whirling  velocity  which  cannot  be  very 
different  in  direction  and  magnitude  from  the  velocity  of  the 
mass  of  water  in  the  volute.  The  vortex-chamber  is  sometimes 
provided  with  guide-blades  following  the  direction  of  free  vortex 
stream-lines  (equiangular  spirals)  so  as  to  prevent  irregular 
motion. 

Centrifugal  pumps  work  under  different  conditions  from 
turbines,  and  hence  there  are  corresponding  differences  neces- 
sary in  their  design.  They  work  best  for  the  particular  lift  for 
which  they  are  designed,  and  any  variation  from  this  lift  causes 
a  rapid  reduction  in  the  efficiency. 


552 


ANALYSIS  OF  CENTRIFUGAL   PUMP. 
FIG.  315. 

UPPER  WATER  LEVEL  - 


FIG.  316. 


ANALYSIS   OF  CENTRIFUGAL   PUMP. 


55S 


2.  Analysis  of  the  Centrifugal  Pump.  —  Designate  the 
velocities,  angles,  and  pressure-heads  at  the  inlet-  and  outlet- 
surfaces  of  the  wheel,  Figs.  315  and  316,  by  the  same  symbols 
as  iri  the  case  of  the  turbine,  Art.  4. 

Let  Q  be  the  delivery  of  the  pump  in  cu.  ft.  per  sec. 

Let  Hg  be  the  gross  lift  including  the  head  equivalent  to  the 
total  hydraulic  resistance  (7/r),  the  actual  lift  (77rt),  and  the 


head    equivalent   to    the    velocity    of  delivery 
Then 


viz.,,—. 


r         .          . 

hr  includes  the  heads  equivalent  to  the  resistance  in  the 
suction-pipe  (^),  in  the  delivery-pipe  (//2),  and  in  the  wheel- 
passages  (7z3),  so  that 


FIG.  317. 


FIG.  318. 


Ha  includes  the  height  of  suction  (ks)  and  the  height  of  delivery 
(k      so  that       * 


554 


LOSSES   IN  HYDRAULIC  RESISTANCE. 


The  height  of  suction  (/^)  is  generally  taken  to  be  the 
vertical  distance  between  the  lower  water-level  and  the  axis 
of  the  pump,  but  this  is  incorrect  and  may  lead  to  serious 
errors.  The  true  height  of  suction,  i.e.,  the  height  to  which 
the  water  must  be  raised  before  the  pump  will  commence  to 
do  work,  should  be  measured  from  the  lower  water-level  to 
the  top  of  the  impellor  or  to  the  top  of  the  wheel  according  as 
the  axis  of  the  pump  is  horizontal  or  vertical. 

The  actual  loss  in  hydraulic  resistances  between  the  suc- 
tion-level and  the  eye  of  the  pump  may  be  determined  by 

the  following  method  suggested  by 
Albert  F.  Hall.  A  long  gauge- 
glass  AB,  with  a  cast-iron  cap  CD, 
is  fitted  into  the  top  of  the  suction- 
pipe.  The  water  rises  in  the  tube 
to  a  certain  level  aa,  and  the  pres- 
sure in  this  tube  can  be  directly 
measured  by  means  of  the  gauge  G. 
If  HB  is  the  barometric  head  and 
HG  the  gauge-reading,  in  feet,  then 
HB  —  HG  is  the  actual  dynamic  head 
at  aa.  Hence  if  Hr  is  the  static 
head,  i.e.,  the  vertical  distance  be- 
tween -  the  suction-level  and  aa, 
(HB  —  //<;)  —  H'  is  the  loss  due  to 
the  several  hydraulic  resistances. 

The  air-chamber  thus  con- 
structed seems  to  cause  a  steadier 
flow  of  water,  and  experiment  shows 
that  the  variation  of  level  at  aa  is 
small  and  is  only  about  \  to  ^  inch. 
In  pumps  which  are  fed  on  both 
sides,  Fig.  313,  the  steadiness  of  the 


FIG.  319. 


level  is  increased  by  placing  an  air-chamber  on  each  suction- 
bend,   connecting   the  two  at  the  upper  end   by  a  horizontal 


ANALYSIS   OF  CENTRIFUGAL   PUMP.  555 

pipe.  A  valve  in  the  middle  of  the  pipe  may  communicate 
with  a  vacuum  pump,  and  each  chamber  may  also  be  controlled 
by  a  separate  valve. 

The  water  apparently  flows  through  the  bend,  past  the 
orifice,  as  over  an  elastic  cushion. 

The  total  work  done  on  the  pump  per  second 


(v,,,"u,  -  vu.'u,)  =  wQH,,,       .     .     .     (i) 

•and  therefore 

V,,."u2  —  VM.'UI  =  gKg  ......      (2) 

The   efficiency    ?/  =  ~  =        „       J*      ,  ....      (3) 

JJ-r/  "«'    U2  Vw'  Ul 

and    gHf/    =    ?;(vt,/'u2  —  Vj/uJ    is   the   fundamental    equation 
governing  the  design  of  a  centrifugal  pump. 

The  water  spreads  out  more  or  less  radially  from  the  eye 
of  the  pump  and,  for  simplicity  of  calculation,  it  is  often 
assumed  that  y  =  90°.  Then 

vwf  =  o,      i\  =  vrf     and 


Again,  eq.   (2)  becomes 

*«/X  =  gHe  -  («2 

which  may  be  written  in  the  form 


a  quadratic  giving 


VgH,       Vg*,     *  gH,        4 

By  means  of  this  result  the  following  Table  has  been  pre- 
pared and  gives  the  values  of —    2      t  corresponding  to  different 

^      I ! 

values  of  —  r        and  B : 


556 


VALUES   OF  BLADE  ANGLES,  ETC. 


7'    " 

ft 

= 

*ffTe 

15° 

165° 

|      30° 

150° 

45° 

135° 

60° 

120° 

75° 

I05° 

90° 

I 

! 

I.O 

3.983 

.251 

2.189 

.457 

.6l8 

.618 

1-330 

•752 

.144 

.876 

i 

•9 

3.633 

.275 

2-047 

.489 

.547 

.647 

•293 

•773 

.127 

.887 

.8 

3.290 

.304 

.909 

•  524 

•477 

.677 

•257 

•795 

.112 

.898 

•7 

2.951 

•339 

•775 

.563 

.409 

.709 

.222 

.818 

.098 

.9IO 

.6 

2.620 

.382 

.647 

.607 

.344 

•744 

.188 

.842 

.083 

.922 

.5 

2.301 

•434 

•  522 

.656 

.281 

.781 

•154 

.866 

.069 

•935 

•  4 

1.992 

.500 

.405 

.712 

.220 

.820 

.121 

.890 

•055 

.948 

•  3 

1.706 

086 

•293 

•  773 

.161 

.861 

.090 

.917 

.041 

.961 

.2 

1.441 

.695 

.188 

.842  | 

.105 

•905 

•059 

•  944 

.027 

•  973 

.1 

1.204 

.830 

.090 

.917 

.051 

•  951 

.029 

.971 

.013 

.987 

Remarks  on  the  Angles  y  and  a,  and  on  the  Curve  of  the 
Blade.  —  The  assumption  of  a  radial  flow  from  the  eye,  i.e., 
that  Y  =  90°,  cannot  of  course  be  true  and,  possibly,  is  not 
even  approximately  accurate,  but  is  solely  made  for  the  pur- 
pose of  securing  simplicity  in  the  calculations.  In  fact,  the 
water  flows  towards  the  eye  with  a  uniform  motion  parallel  to 
the  axis  of  rotation,  while  at  every  point  between  the  inlet 
and  outlet  of  the  wheel  the  motion  of  a  fluid  particle  is  the 
resultant  of  a  constant  angular  acceleration  (Art.  2  1  ,  Chap. 
I)  and  of  a  radial  acceleration  due  to  centrifugal  force,  viz., 


At  the  inlet  the  tip  angle  a  may  be  varied  between  wide 
limits,  but  its  value  should  be  such  as  to  make  the  efficiency 
as  great  as  possible:  But  a  cannot  be  expressed  as  a  function 
of  the  mechanical  and  hydraulic  resistances,  and  it  is  therefore 
impossible  to  find,  analytically,  the  value  of  a  which  will  make 
these  resistances  a  minimum.  The  best  value  for  a  can  only 
be  determined  by  a  very  extensive  series  of  experiments. 
According  to  the  best  practice,  however,  a'  usually  lies  between 
50°  and  60°. 

The  curve  of  the  blade  is  in  some  cases  a  circular  arc.  In 
many  first-class  wheels  it  is  a  cycloid  developed  by  a  circle  of 
a  diameter  equal  to  one  fourth  of  the  diameter  of  the  wheel  v 
the  cycloidal  arc,  at  the  innermost  point,  being  tangential  to 


ANALYSIS  OF  CENTRIFUGAL   PUMP.  557 

the  hub.  Brix,  again,  has  deduced  the  following  formula, 
giving  the  angle  0  between  the  blade  and  the  direction  of 
rotation  at  any  point  distant  r  from  the  axis: 


COt  0  = 


Ir-.-^ 

V       -    G>* 


Q     }  <*>*         co    }  I  2nr 


where  2nrdY  =  (2nr  —  nt)d',  T=  work  of  pump  to  radius  r\ 
ri  =  radius  to  inner  end  of  blade;  vr  =  radial  velocity  at  inlet; 
n  =  number  of  blades  ;  t  =  thickness  of  blade. 

By  plotting  the  values  of  0  corresponding  to  different  values 
of  r  the  curve  of  the  blade  may  be  defined. 

It  is  essential  that  there  should  be  no  dissipation  of  energy 
in  eddy  motion  at  the  inlet,  and  the  direction  of  the  relative 
velocity,  Vl  ,  should  therefore  be  tangential  to  the  blade-tip 
at  a,  Fig.  315.  Then, 

from  the  triangle  adc,   V*  =  v*  -)-  u^  —  2vlul  cos  y,        (5) 
•  '       fkh,   V*  =  v*  +  u?  -  2v  ^  cos  tf,        (6) 

and  therefore 

V?  -  -  F22       z/22  —  u?       \y*  —  v*    _  v2u2  cos  d      T/I?/I  cos  y 

.  ig       *g      ~^g~      g       ~ir    7) 

The  water  leaves  the  wheel  with  a  velocity  v2  ,  and  carries 

v  2 
away;  in  its  energy  of  motion,  viz.,  —  ,  an  important;  portion 

of  the  work  done  on  the  pump  by  the  prime  mover.  If  the 
whole  of  this  energy  could  be  made  available  for  increasing 
the  pumping  power,  then,  by  Bernoulli's  theorem, 


i  (  . 

2g  W          2g  W          2g 

Also, 


y   2  __    U   2 

the  term     2  —  —  L  being  the  variation  of  pressure-head  due  to 


558 

centrifugal  action  between  the  \vheel  inlet  and  outlet.     Hence,, 
by  eqs.  (8)  and  (9), 

7,  2  T72  772  ?/  2  _         2  7,  2  _   7,  2 

Ar  +  ff,+    ±-=  ^  -      -2 +  ^-  -'-  +  -'-  -L 

a     '        ?  cr  -7  <r  ?  <r  2(T 

o  "&  "<S  ~o 

_   T^COS  d  _      7W  COSJK 
£• 

where  //r  =  //L  -|-  //2  -f-  h^  =  the  head  equivalent  to  the  total 
hydraulic  resistances. 

If  the  inle^  flow  is  radial,  i.e.,  if  ;/ =  90°,  and  if  the 
hydraulic  resistances  and  the  velocity  of  delivery  can  be 

diminished  to  such  an  extent  that  hr  and  --  become  sufficiently 

o 

small    to    be    disregarded    without    much    error,  then    eq.  (10) 
becomes 

H  =jaic^  =  !!!/x 

g         g 

and  the  total  available  energy  is  transformed  into  useful  work. 

Volute. — The  water  issues  from  the  outlet-surface  into  a 
casing,  or  volute,  which  surrounds  the  wheel  and  which  should 
always  be  designed  in  such  a  manner  that  the  disturbance  in 
the  fluid  mass  might  be  as  small  as  possible,  since  the  least  dis- 
turbance in  the  stream-line  motion  causes  a  loss  of  energy  in 
shock.  -Thus  its  sectional  area  on  any  normal  plane,  through 
the  centre  of  the  wheel,  should  be  proportional  to  the  quantity 
of  water  which  flows  across  the  section  in  the  same  given  time, 
and  the  corresponding  mean  velocity  of  flow,  vs,  in  the  volute 
is  necessarily  constant.  If  the  width  of  the  volute  is  also  con- 
stant, its  profile  will  evidently  be  an  Archimedean  spiral.  By 
making  the  gradually  increasing  sections  sufficiently  Irrge  the 
velocity  of  flow,  7^,  may  be  made  very  small,  and  a  bell-mouth 
entrance  into  the  discharge-pipe  may  become  unnecessary. 

Figs.  320  to  325  are  of  interest  as  showing  the  experimen- 
tal stages  through  which  Farcot's  pump  passed  in  the  process, 
of  its  gradual  development. 


AN 'A LYSIS   OF  CENTRIFUGAL   PUMP. 


559 


The  assumption  that  the  total  available  energy  may  be 
transformed  into  useful  work  is  altogether  inadmissible  in 
practice,  as  a  large  portion  is  consumed  in  overcoming  frictional 
resistance  and  in  the  production  of  eddies. 


FIG.  320. 


FIG.  324. 


FIG.  322.  FIG.  325. 

Again,  even  with  the  most  perfectly  designed  volute,  the 

7/2 

hypothesis  that  the  whole   of  the  energy  of  motion,     2  ,   may 

O 

be  utilized  in  increasing  the  pumping  power  is  untenable. 
The  water,  as  it  leaves  the  wheel,  with  a  velocity  7»2 ,  impinges 
upon  the  fluid  mass  in  the  volute,  and  the  radial  component 
vr"  of  ?'2  must  necessarily  be  almost,  if  not  wholly,  destroyed, 

the  corresponding  loss  of  head  being  — — .  The  tangential 
component  of  ?'2,  viz.,  vm",  is  also  changed  into  vst  the  velocity 


560  MAXIMUM  EFFICIENCY   OF  CENTRIFUGAL   PUMP. 

of  flow  in  the  volute,  and,  if  the  change  were  gradual,  there 

would  be  a  gain  of  head  equal  to — ,  but,  as  the  change 

(v.  tt v  \2 

is  abrupt,  there  is  a  loss  of  head  in  shock  equal  to  —^ ^.. 

Hence  the  net  gain  of  head  available  for  increasing  the  pump- 
ing power 

//o  ,.   2  /,-,      f!  f.    \2 

_       tt<  *-  s  \*  it)  Si 

~^2g    "   2g~  2g 

V'YV"""     "V-- (12) 


g 

which  is  a  maximum  and 


V,2 


= 


V5(VW"  —  7/  )  7'  2 

The  term  -  -  should  be  substituted  for  -  '-   in  eq. 

(8),  and  then 

^i  +  ^»+~  +  ~H 1~  =  — ^2  +  ^+'  •       (T3) 

Hence,    by   eqs.    (7)    and    (13),  the    following  equation  is 
obtained  instead  of  eq.  (10): 

1) .  i>  (v    "    —  1'  )  V  U    COS   $  7^  //     COS  V  V  2 


a 

u~~  v      vu  cos 


and  therefore 

II     * \/      *  TT     I  IT  TT     \  T7    ,*  XT'    11        /^/\O     ir 

TT  U2  V2       I      V*VV^       —     Vs)  V««  *  V1U1  COS  7X  ,         N 

H«=   —^-         ~Y~      '^~'K~      t        (I4) 

r.,     2  /     ,       / 

Maximum  Efficiency. — If  the  terms  hr,  — ,  and  -J^-!L 


6  "  o 


MAXIMUM  EFFICIENCY  OF  CENTRIFUGAL   PUMP,  561 

are  sufficiently  small,  as  compared  with  Ha  ,  to  be  disregarded 
without  much  error,  and  if  y  =  90°,  then  eq.  (14)  becomes 

Ha  =  ^^ (15) 

But 

sin  tf  sin  0  cos  6 

=  va  cos  8  = 


9  —       9 /JQ     ,      Jv*  w                   2                     —      2             /ft     I            ' 

2  sin  (ft  +  tf)  2  sin  (/?  +  rf) 
Therefore 

sin2  0  2sin/?sin(/?  +  2d) 


,gHa~u*\\  — -T- 

""  \^~r  uj '  3111~  {r  ~r  u; 

(16) 


sm8(/?+<5)J  sin2 

or 


2gHa 


ffi  .  n 

and  the   efficiency  ?;  =  -—77—  =  - 


2  cos  tf.sin 


or  77  =  -{i  +  tan  tf  cot  (/?+  rf)[. 

The  efficiency  increases  as  /?,  the  outlet-tip  angle,  dimin- 
ishes, and  would  be  unity,  i.e.,  perfect,  if  ft  could  be  O. 

If  the  blade  is  radial  at  the  outlet,  i.e.,  if  fi  —  90°,  then 

the  efficiency  —  -J  I  +  tan  6  cot  (90°  +  <?)} 


and  could  never  exceed  £. 

For  any  given  value  of  /3  the  efficiency  is  a  maximum  when 


562  MAXIMUM  EFFICIENCY  OF  CENTRIFUGAL   PUMP. 

This  can  be  easily  shown  analytically,    or  geometrically  as 
follows : 

Upon  any  line  AB  as  diameter  describe  a  semicircle.     Draw 

a  chord  *AC  making  the  angle  ft 
with  AB.  Draw  any  chord  AD 
making  an  angle  d  with  A  C,  and 
join  DB  intersecting  A  C  in  O. 

The  efficiency  is  greatest  when 

% x"-4    tan  #  cot  (ft  ~\-  (5)  has   its  greatest 

3~~  B  value. 

FlG>  326>  But  since  the  angle  in  a  semi- 

circle is  a  right  angle, 

DO      AD       DO 
tan  $  cot  (04-  0}  - 


and  the  efficiency  is  therefore  greatest  when          is  a  maximum. 


—  is  nil  both  when  D  coincides  with  A  and  also  with 
DB 

C,  and  must  consequently  be  a  maximum,  or  stationary,  at 
some  position  of  Z>  between  A  and  C.  This  position  is  at  once 
found  from  the  condition  that  if  Dl  is  a  consecutive  point  and 
if  D^B  is  joined  intersecting  A  C  in  Ol  ,  then 


_ 
'DB  ~~~  T\B' 

so  that  DDl  must  be  parallel  to  OOl  or  ^  C,  and  is  therefore 
a  tangent  to  the  semicircle  at  Dy  which  is  necessarily  the  middle 
point  of  the  arc  AC.  Hence,  since  the  arc  AD  —  the  arc  CD, 

the  angle  ABD  =  the  angle 

and  therefore 

90°  -  (ft  +  6)  =  <* 
or  /?  +  2d-9o°. 


MAXIMUM  EFFICIENCY  OF  CENTRIFUGAL   PUMP.  563 

Hence,  too, 
the  max.  efficiency  —  I  -\-  tan  tf  cot  (90°  —  6) 

=  1  «*•*  =  £  sec*  (4S- - 

The  outlet  velocities  corresponding  to  this  maximum  effi- 
ciency are  represented  by  the  sides  of  the  triangle  fkh,  Fig. 
327.  The  two  triangles  fnh  and  fxh  are  equal  in  every  respect. 


Also, 


FIG.  327. 

2  sin2  <T=  i  —  cos  2d  =  i  —  sin  ft 
2  cos2  d  —  I  -f-  cos  2d  =  i  -j-  sin  ft. 


Hence 


and 


•or 


sn 


sin  (90°  + 


=  «  tan  *  = 


2  sin  /? 


l  — 


I  -4-  sin  G  ' 


I  -|-  sin 
2  sin  / 


564  MAXIMUM  EFFICIENCY  OF  CENTRIFUGAL   PUMP. 

sin  ft 


Als°'   "  =  ^ 


sn 
cos 


/sin  tf  (i+sintf)      rr 
=  u2sm/3=*/  -         L-IL       -V#«, 


.      ,  /sin  /?  (I  -  sin  /?) 

and         V      ==  v2  sm  tf  =  A  /  -          -^—        —2gHa. 

From  these  equations  the  following  Table  has  been  pre- 
pared : 


"1 

TI 

V 

/3 

s 

*^H~a 

V^a 

**gHa 

17 

5° 

42°  30' 

2.497 

.296 

.20 

•92 

8  26' 

40  47 

•  977 

.383 

•25 

•  87 

10 

40 

.838 

.416 

.267 

.80 

15 

37  30 

•  56 

•  500 

•30 

•79 

20 

35 

.40 

•58 

•33 

•74 

30 

30 

.22 

.70 

•35 

.67 

40 

25 

•13 

.80 

•34 

.61 

45 

22  30 

.09 

.84 

•32 

•58 

50 

20 

•073 

.87 

•30 

•  56 

58  36 

15  42 

.041 

.92 

•  25 

•54 

60 

15 

.038 

•93 

.24 

•  53 

i 

The  value  of  the  radial  component,  rr",  is  greatest  when 
sin  /3(i  —  sin  /?)  is  a  maximum,  i.e.,  when  sin  ft  =  ^  or 
ft  =  30°  --=  6. 

Two  values  of  ft,  the  one  less  and  the  other  greater  than 
30°,  correspond  to  every  other  value  of  vr". 


Generally  speaking,  vr"  lies  between  \^2,gHa  and 
and  the  assumption  is  sometimes  made  that  the  radial  com- 
ponent of  the  velocity  of  the  water  as  it  passes  through  the 
wheel  is  constant  and  equal  to  \ 


EXAMPLE.  —  A  centrifugal  pump,  with  an  outlet-tip  angle 
(/?)  of  20°,  has  an  efficiency  of  60  per  cent.  Assuming 
vr"  =  \  \/2gHa  ,  then 


WHIRLPOOL-CHAMBER.  565 

gHa  =  .6u2vw"  =  .6«2(«a  -  *v"  cot  20°) 

=  .6«y«2  -  -  1/2^x2.7475), 


4 

and  #2  =  1.32  \/2gHa. 

Also, 

i     

J7  =  vr"  cosec  20°  =  -  V2gHa  X  2.9238 
4 


X  .73095- 
Therefore 


Hence,  if  the  inlet  flow  is  radial,  equation  (14)  gives 


?y/2costf 


and 


Certain   existing   experimental   results   give    .^2Ha   as   an 
average  value  of  hr\  and  taking  .o$Ha  as  the  average  value  of 


then 


+  .03/4  -  .208^,  =  .242/4. 


The  term  ~  —  -          —  ^-  must  necessarily  vary  considerably 

with  the  design  of  the  pump. 

3.  Thomson's  Vortex  or  Whirlpool-chamber  (Figs.  328 
and  329).  —  It  has  been  suggested  that  the  energy  of  motion 
inherent  in  the  water,  as  it  leaves  the  wheel,  may  be  more 
completely  utilized,  and  the  pumping  power  therefore  in- 
creased, by  the  addition  of  an  exterior  chamber  of  radius  r3  , 
in  which  the  water,  in  virtue  of  its  motion,  is  left  free  to 


566 


WHIRLPOOL-CHAMBER. 


revolve,  and  tends  to  assume  the  condition  designated  by 
James  Thomson  as  the  vortex,  or  whirlpool,  of  free  mobility. 
The  centrifugal  action  of  this  fluid  mass  develops  an  outward 
force  which  is  added  to  the  outward  force  developed  within  the 
wheel  and  materially  increases  the  pumping  power.  The  out- 
ward force  produced  within  the  wheel  is  due  to  centrifugal 


FIG.  328. 


FIG.  329. 


action  only,  if  the  blades  are  radial ;  but  if,  as  is  generally  the 
case,  the  blades  are  curved,  it  is  partly  due  to  the  radial  com- 
ponent of  the  pressure  between  the  blades  and  the  water,  and 
this  pressure  may  be  very  great  if  the  pump  is  run  at  a  high 
speed. 

The  chief  properties  characterizing  the  fluid  mass  in  the 
whirlpool-chamber  are  the  following : 

(i)  Each  fluid  particle  moves  with  a  velocity  (V)  inversely 
proportional  to  its  distance  (r)  from  the  axis  of  rotation.  Thus 


vz  being  the  water's  velocity  at  the  outlet-surface  of  the  whirl- 
pool-chamber. 

(2)  The  angle  (0)  between  the  radial  distance  (r)  to  any 
particle  and  its  direction  of  motion  is  constant,  and  the  stream- 
lines are  therefore  equiangular  spirals. 

Thus  if  vr'"  and  vw'"  are  the  radial  and  tangential  com- 
ponents of  vz , 


—  Vz  COS 

—  t/«  sin 


=  7;   cos 


WHIRLPOOL-CHAMBER.  567 

and  therefore 

^L  -  vj  =  i»  -  V^L 

vj"   -*.       r,       v.'" 

(3)  Each  particle  is  free  to  move  to  any  position  within  the 
whirlpool  without  interfering  with  the  general  motion  of  the 
other  particles,  as,  in   moving  towards  or  from  the  centre,  it 
assumes  of  itself,  subject  simply  to  the  laws  of  motion  under  a 
central  force,  the  velocity  due  to  its  position  in  the  whirlpool. 

(4)  For  any  equal  particles,  whatever  positions  they  may 
momentarily  occupy  in  the  whirlpool,  the  sum  of  the  energies 
corresponding  to  velocity,,  to  pressure,  and  to  height  is  con- 
stant. 

Thus  each  particle  gives  up  its  velocity  in  accordance  with- 
the  law  of  motion  just  stated,  and  the  head  available  for 
increasing  the  pumping  power 

-  VJ.  _  !^  =  vi(l   _  rJ\ 
*S      tg  "      \         r) 


Again,   the    term   --^  ---  -,   representing  the  gain  of 

<s 

head  in  passing  from  the   whirlpool-chamber  into  the  volute, 

must  be  substituted  for  the  term  ^^  -  —  .      Thus 

g 

TT 

the  efficiency  =  -=£, 
He 

in  which  Hg  =  ujvw"  =  u2(u2  —  V2  cos  ft),  and  the  actual  lift 
is 

'         '  (v«.///-v.)  .        V      h 

--  h- 


Ex.  Assume  that  the  four  last  terms  in  the  preceding  equation  are 
sufficiently  small  to  be  disregarded.     Then 

«22    _    VJ    +    (vr"*    +    Vw''2)(l    -   ~\ 

the  efficiency  =  ' 


—    V*  COS 


568 


WHIRLPOOL-CHAMBER. 


First.  Let  ft  =  90°,  i.e.,  let  the  blade  outlet-lip  be  radial.     Then 
Therefore 

K,_>V  +  (KJ,  +  F4_-;) 

the  efficiency  =  —  — ^ ^_Z 


Second.   Let  /S  =  o,  i.e.,  let  the  blade  outlet-lip  be  tangential.     Then 

7yw"  =  #2  —  Fa     and     z/r"  =  o. 
Therefore 


the  efficiency  = 


The  difference  between  these  two  efficiencies  is  compara- 
tively small  and  diminishes  as  the  diameter  of  the  whirlpool- 
chamber  increases.  Hence  their  values  are  not  largely  influ- 
enced by  the  angle  ft. 

The  above  hypothetical    theory  seems  to  indicate  that  a 


FIG.  330. 

whirlpool-chamber  adds  to  the  efficiency  of  a  centrifugal  pump. 
Opinions,  however,  differ  widely  as  to  the  real  character  of  the 


PRACTICAL   COEFFICIENTS.  569 

flow  of  the  water  within  the  pump,  and  as  to  the  loss  of  energy 
in  shock  on  entering  the  whirlpool-chamber  or  the  volute. 
Some  eminent  authorities  advocate  a  gradually  diminishing 
section,  Fig.  330,  on  the  ground  that  it  tends  to  produce  a 
steadier  action,  while  other  authorities,  equally  eminent,  claim 
that  a  flaring  vortex  tends  to  increase  the  efficiency,  and  it  is 
urged  that  by  widening  the  chamber  from  the  depth  at  the 
wheel-outlet  to  a  much  greater  depth  at  its  exterior  surface, 
the  water  will  lose  its  energy  of  motion  much  more  rapidly  and 
will  leave  the  chamber  with  a  velocity  more  nearly  equal  to 
that  in  the  discharge-pipe. 

Experiments  are  urgently  needed  to  throw  light  upon  this 
important  subject. 

4.  Practical  Values.  —Let  dl ,  d2  be  the  depths  of  the  inlet- 
.and  outlet-surfaces. 

Let  tv ,  t2  be  the  blade  thickness  at  inlet  and  outlet. 

Let  n  be  the  number  of  blades. 

Let  the  inlet  area  =  sectional  area  of  supply-pipe. 

Then,  if  y  —  90° 

(2,nr^  —  ntl  cosec  o^d^u^  =  --  Q  =  nr*vr' 

=  (2nr2  —  ;z/2  cosec 

10 
the  coefficient  —  being  an  average  value  and  depending  upon 

practical  considerations. 

The  following  values  are  sometimes  adopted  in  practice: 


n  =  4  to  10; 

tl  =  t2  —  .2  in.  to  .625  in.  ; 


d2  =  dl  or  =  %dl  ,  according   as  the  pump- 
faces  are  parallel  or  coned. 


57°  EXAMPLE 

Ex.  The  hypothetical  advantage  of  a  whirlpool-chamber  may  be 
observed  by  a  consideration  of  the  comparative  efficiencies  of  two 
pumps,  which  are  precisely  similar  in  every  respect  excepting  that  one 
has  a  whirlpool-chamber  of  62  ins.  diameter.  Each  pump  delivers  20 
cu.  ft.  of  water  at  a  speed  of  225  revolutions  per  minute.  The  diameters 
of  the  suction-  and  discharge-pipes  =  20  ins. ;  the  diameter  of  the 
wheel  =  36  ins.  ;  the  depth  of  the  whirlpool-chamber  =  the  depth  of  the 
wheel  at  outlet  =  5!  ins. ;  y  —  90° ;  /3  =  42°  20'  22". 65;  number  of  wheel- 
blades  =  6  ;  thickness  of  blade  =  £  in. 

The  actual  lift  is  given  by 


-  v.) 


„ 

or  //a  = 


or  2P° 

according  as  the  pump  has  not  or  has  a  whirlpool-chamber. 

225  x  it  x   3 
«a  =  ~        f^        -  =  3  SIT  ft.  per  sec.; 


cosec  (3  =  1.484; 

144  x  20 


=  5.523  ft.  per  sec.; 


x36-6x|x   1,484 
V<i  =  vr''  cosec  ft  =  8.196  ft.  per  sec.; 
-  EV  _  591-473. 


cot  /?  =  i  .097 ; 
vw"  =  «a  —  «v"  cot  /5  =  29.298  ft.  per  sec.; 

20 

Vd  =  v*=  -TTTa  =  9 A  ft.  per  sec.; 
irfi. 


v,(vw"  —  vj       184.508 

— - —        = =  gam  of  head  in  passing  from  wheel  into  volute; 

vf      4 1 .986 


2£"  g 

•u*  =  vj"1  +  vw"*  =  888.876; 
vS(         rS\  _  888. 876 /         i82\  _  294.596 

l  ~ -          1  --  ~- 


EXAMPLE.  571 


i8 
v^     =  —  x  29.298  =  17.012  ft.  per  sec.; 

—  vs)  _  71-919. 


w*    =  1035.; 

Hence  for  the  pump  without  a  whirlpool-chamber 

ff  —  591-473  +  184.508  _  41.986  _ 

_  733-995  _ 

g 

or  gHa  —  733.995  —ghr, 

and 


the  efficiency  =  =  "         =  .708  - 

«aVw  1035.889  1035.889 


For  the  pump  w/M  a  whirlpool-chamber 


294.596        71-919       41 
__.+__       - 


=  916.002 

o 
Or  ^"^a   =   916.002    — 


and  the  efficiency  =  91-OM-^,  _  __ 

1035.889  1035.889' 

which  is  considerably  greater  than  the  first  efficiency. 


572  EXAMPLES. 


EXAMPLES. 

1.  Find  the  H.P.  required  to  drive   a  centrifugal  pump   of   14   ft. 
diameter,  and  with  radial  vanes,  making  60  revolutions  per  minute  and 
delivering  900,000  gallons  of  water  per  hour.     If  the  lift  is  30^  ft.  find 
the  efficiency.     Assume  that  the  water  on  entering  has  no  velocity  of 
whirl.  Ans.   27.5  ;  .5. 

2.  The  wheel  of  a  centrifugal  pump  is  .6  ft.  in  diameter  ;  the  turning 
moment  on  the  spindle  is  12  Ibs.-ft.     If  160  gallons  of  water  are  raised 
per  minute,  find  the   mean  velocity  with  which   the  water  leaves  the 
wheel ;  assuming  that  on  entering  it  has  no  velocity  of  whirl. 

Ans.  24.  i  ft.  per  sec. 

3.  A  centrifugal  pump  has  a  36-in.  wheel   of  a  uniform  breadth   of 
5^  ins.     The  wheel  makes  225  revolutions  per  minute  and  delivers  20  cu. 
ft.  of  water  per  second  into  a  discharge-pipe  of  20  ins.  diameter.     The 
angle  (ft)  of  the  blades  at  the  outer  periphery  is  42°  20'.     Assuming  the 
velocity  of  discharge  to  be  the  same  as  the  mean  velocity  of  flow  in  the 
volute  and  disregarding  vane-thickness,  find  (a)  the  peripheral  speed  ; 
(b)  the  velocity  of  whirl  and  radial  velocity  of  flow ;  (c)  the  gain  of  head 
available  for  useful  work  on  entering  the  volute,  and  (d}  the  efficiency. 

There  are  six -|-in.  blades.  If  a  62  in.  whirlpool-chamber  is  added,  find 
the  gain  of  head  available  for  useful  work,  (e)  due  to  chamber;  (/)  on  enter- 
ing volute. 

Ans.  (a)  33.35  ft.  per  sec.;  (£)  29.425  and  5.4  ft.  per  sec.;  Or)  5.8 
ft.;  (d)  .708;  (e)  9.27  ft.;  (/)  2.27  ft. 

4.  A  centrifugal  pump  with  a   i2-in.   fan  delivers  1000  gallons  per 
minute,  the  actual  lift  being  20  ft.  and  ihegross  lift  (allowing  for  friction, 

etc.)  30  ft.    Find  the  revolutions  of  the  pump  per  minute  lvw"  =  —  J. 

Ans.  836.52. 

5.  In  a  centrifugal  pump  the  external  diameter  of  the  fan  is  2  ft.,  the 
internal  i  ft.,  and  the  depth  6  in.     Determine  the  speed  and  efficiency 
of  the  pump  when  delivering  2000  cu.  ft.  per  minute  against  a  pressure 
head  of  64  ft.,  the  inclination  of  the  wheel-vanes  at  outlet-surface  being 
90°,  and  y  being  also  90°.  Ans.  619.24  revols.  per  min.;  .4866. 

6.  A  centrifugal  pump  delivers  1500  gallons  per  minute.     Fan,  16  in. 
diameter;   lift,  25    ft.;    inclination  of  vanes   at  outer  periphery  to   the 
tangent,    30°.      Find   the   breadth   at   the   outer    periphery,    and  .  a,lso 
the  revolutions  per  minute,  assuming  the  gross  lift  to  be  i|  times  the 

actual  lift,  and  that  z>TO"  =  — . 


EXAMPLES.  575 

Also  find  the  proper  sectional  area  of  the  chamber  surrounding  the 
fan  for  the  proposed  delivery  and  lift.  Examine  the  working  of  the 
pump  at  a  lift  of  15  ft.  (vw"  =  o). 

Ans.  Breadth,  $  in.  ;  revolutions,  700  ;  23.5  sq.  ins. 

7.  For  a  given  discharge  (0  and  head  (//),  and  considering  only  the 
losses  of  head  due  to  flow  and  to  the  resistance  in  the  wheel,  show  that 
the  maximum  efficiency  of  a  centrifugal  pump  of  diameter  D  is 


i  -A 


Q    ' 


A  being  a  constant  depending  on  the  size  of  the  wheel. 

8.  A  centrifugal  pump  with  an  efficiency  of  .75  and  a  radial  flow  at 
inlet,  lifts  35  cu.  ft  of  water  per  second  a  height  of  20  ft.     At  the  outer 
periphery  the  vane-angle   (ft)  is   15°  and   the  radial  velocity  is  5  ft.  per 
second.     If  the   wheel   makes    140   revolutions   per  minute,  find  (a)  its 
diameter.     If  the  diameter  of  the  outer  periphery  of  the  wheel  is  three 
times  that  of  the  inner  periphery  and  if  the  radial  velocity  at  the  latter 
is  8  ft.  per  second,  find  (b)  the  vane-angle  at  the  inner  periphery  and  (c) 
the  depths  of  the  wheel  at  the  inner  and  outer  peripheries. 

Ans.   (a)  5.455  ft.  ;  (b)  30°  58' ;  (c)   .765  ft.  ;  .41  ft. 

9.  The  pump  in  the   preceding   example  is  supplied  with  a  vortex- 
chamber  of  6£  ft.  diameter.     Show  that  the  "gain  of  head  "  is  a  maxi- 
mum when  the  velocity  of  flow  in  the  volute  is  8.46  ft.  per  second.    Also 
show  that  the  frictional  loss  of  head  is  4.18575  ft. 

10.  In    a   centrifugal    pump   the   diameter  of  the  fan  =  12  ins.,  the 
depth  =  2  ins.,  the  lift  =  25  ft.,  and  the  delivery  =  300  cu.  ft.  per  minute. 
Determine  (a}  the  speed  ;  (ff)  the  efficiency ;  and  (c)  the  power  expended 
when  the  vane-angle  (ft)  at  the  outer  periphery  is  (i)  90° ;  (2)  45° ;  and 
(3)  30° ;  Y  being  90°. 

Ans.  (i)  (a)  785      revols.  per  min. ;  (b)  .47  ;  (c)  30      H.P. ; 

(2)  (a)  805.8      (6)  .58  ;  (c)  24.4  H.P. ; 

(3)  (a)  846.1      "         "        "         (b)  .68  ;  (c)  22.9  H.P. 

11.  A   centrifugal  pump  delivers    10,000  gallons  per   minute.     The 
actual  lift  is  50  ft.    The  radial  velocity  at  the  outlet-surface  is  one  eighth 
of  that  due  to  the  actual  lift  and  ut  =  2vw".     Find  (a)  the  radius  of  the 
wheel ;  (b)  the   vane-angles  ;  (c)  the   speed  of  the  wheel ;    (d)  the   effi- 
ciency, taking  y  =  90° ;  and  d*.  =  d*  —  -£. 

Ans.   (a)  1.9  ft.;  (b)  56°  16';  23°  16';  (c)  331  revols  per  min. ;  (d)  .74. 

12.  The  internal  and  external  diameters  of  the  fan  of  a  centrifugal 
pump  with  radial  flow  at  inlet  are  9  ins.  and  18  ins.,  respectively;  the 
depth  is  6  ins.,  and  it  passes  400  cu.  ft.  per  minute  against  a  pressure 
head  of   16  ft.     The  inclination  (ft)   of  the  discharging-lips  of  the   fan 
being  30°,  determine  (a)  the  speed  ;  (b)  the  efficiency  ;  (c)  the  power  ex- 


574  EXAMPLES. 

pendecl ;  and  (</)  the  inclination  of  the  receiving-lips  of  the  fan.  Find 
(e)  the  efficiency  when  a  whirlpool-chamber  of  36  ins.  diameter  sur- 
rounds the  fan. 

Ans.  (a)  413.58   revols.    per  min. ;  (b)   .571  ;  (c)  21.23  H.P.  ;  (d) 
19°  12';  (e)  .581. 

13.  The  lift  of  a  centrifugal  pump  is  24$  ft.     The  efficiency  of  the 
pump  is  .75,  and  the  radial  velocity  of  flow  at  outlet-surface  of  fan  is  5  ft. 
per  second.     If  cot  y  —  4,  find  the  peripheral  speed  of  the  fan. 

Also  find  its  diameter,  if  the  fan  makes  160  revolutions  per  minute 
•(vwf  =  o).  Find  the  loss  of  head  in  hydraulic  friction. 

Ans.  44  ft.  per  sec.  ;  5^  ft.  ;  3ff  ft. 

14.  The  reciprocal  of  the  efficiency  of  a  C.  P.  is  1.61,  the  peripheral 
(MI)  and  radial   (vr")  velocities  at   outlet  are  35   and  9  ft.   per  second 
respectively.      Find  the  lift  and  the  vane-angle  (/?)  at  outlet. 

Ans.  15!  ft.;  tan'1  f. 

15.  A  centrifugal  pump  with  a  gross  lift  of  17  ft.  delivers  25  cu.  ft.  of 
water  per  second.     At  the  outer  periphery  the  vane-angle  is  80°  and  the 
radial  velocity  is  5  ft.  per  second.    The  diameters  of  the  outer  and  inner 
peripheries  of  the  disc  are  54  ins.  and  18  ins.  respectively,  and  the  hy- 
draulic efficiency  is  .75.     Find  (a)  the  speed  of  the  fan;  (b)  the  vane- 
angle  at' the  inlet  periphery  ;  (c)  the  velocity  of  whirl  at  the  outlet ;  (d} 
the  diameter  of  the  volute  ;  (e)  the  diameter  of  the  suction-pipe. 

If  there  are  six  ^-in.  vanes,  find  (f)  the  width  of  the  disc  at  the  outer 
and  inner  peripheries. 

Assuming  the  velocity  of  flow  in  the  discharge-pipe  to  be  4  ft.  per 
second,  show  that  there  is  a  loss  of  5.026  ft.  of  head  due  to  hydraulic 
friction. 

Ans.  (a)  116  revolutions  per  minute  ;  (b)  41°  14';  (c)  26.49  ft.  Per 
second  ;  (d)  1.094  ft.;  (e]  33.8  in.  ;  (/)  9.64  ins.  ;  4.8  ins. 

16.  The  vane  of  a  centrifugal  pump  or  turbine  is  the  involute  of  a 
circle  concentric  with  the  pump  circumference.     Show  that  Vi  =  Vi  in 

an  I.  F.  or  O.  F.,  and  -77  =  —  in  an  A.  F. 

F2        r* 

17.  If  the  lips  of  the  pump-vanes  are  radial,  show  that  the  efficiency 
cannot  exceed  .5,  but  that  it  might  be  increased  to  .875  by  the  addition 
of  a  whirlpool-chamber. 

18.  A  centrifugal  pump  with  a  2i-in.  fan  pumps  110  1/3  cu.  ft.  per 
second  to  a  height  of  31 J  ft.     The  outlet-lip  makes  an  angle  of  60°  with 
the  periphery.    The  depth  of  the  fan  is  6  ins.    Find  the  peripheral  speed, 
the  H.P.  and  the  speed  of  the  pump  in  revols.  per  minute. 

Also  find  the  loss  of  head  due  to  frictional  resistance. 

Ans.  60  ft.  per  second  ;  1623!  H.P.;  654T6T ;  31 J  ft. 

19.  A  centrifugal  pump,  with  six  ^-in.  blades,  makes  140  revolutions 
per  minute  and  raises  5062^  tons  of  water  per  hour  to  the  height  of  20 
feet.  The  blade-angle  and  radial  velocity  of  flow  at  outlet  are  cot"1  4  and 


EXAMPLES.  575 

5  ft.  per  second,  respectively,  and  the  hydraulic  efficiency  of  the  pump 
is  a  little  more  than  60  per  cent  (=  f  $)•  The  wheel  is  surrounded  by  a 
vortex-chamber  having  a  diameter  20  per  cent  greater  than  that  of  the 
wheel.  Assuming  that  the  inlet-flow  is  radial,  and  that  2vs  =  vw'",  and 
disregarding  frictional  resistances,  determine  the  peripheral  speed,  diam- 
eter and  breadth  of  the  wheel,  and  the  gains  of  energy  in  ft.-lbs.  in  the 
vortex-chamber  and  in  the  volute. 

Ans.  44  ft.  per  sec.;  6  ft.;  8.17  ins;  8070,  8789. 

20.  Compare  the  efficiencies  of  two   centrifugal  pumps,  which  are 
precisely  similar  in  every  respect,  excepting  that  one  has  a  whirlpool 
chamber  of  48  ins.  diameter.     Each  pump  delivers  20  cu.  ft.  per  second 
at  a  speed  of  225  revolutions  per  minute.     The  diameters  of  the  dis- 
charge- and  suction-pipes  =  20  ins. ;  the  diameter  of  the  wheel  =  36  ins.; 
the  depth  of  the  wheel  and  the  whirlpo'ol-chamber  at  outlet  =  3$  ins. ; 
y  =  90° ;   ft  =  22°    38' ;    the    number  of    wheel-blades  =  6  ;  the  blade- 
thickness  =  f  in. ;  hr  —  >^Ha. 

eh 

Ans,  .73  —  A  and  .79  —  A  where  A  =  •&—£-, . 

503.6 

21.  In  a  centrifugal    pump  the  diameters  of  the  suction-  and  dis- 
charge-pipes =  48  ins.;   the  number   of  wheel-blades  =  6;    the  blade 
thickness  =  $  in.;  the  radial  velocity  of  flow  at  outlet  =  2.877  ft.  per 
second  ;  the  velocity  of  flow  in  the  volute  and  discharge-pipe  —  5.817 
ft.    per   second;   the  peripheral    speed    of   the    wheel  outlet-surface  = 
34.6276    ft.   per   second.      Disregarding    the    frictional    losses    in    the 
suction-  and  discharge-pipes  and  in  the  wheel-passages,  determine  the 
velocity  of  whirl  at  outlet,  the  blade-tip  angles  at  outlet,  the  delivery 
in  cubic  feet  per  second,  the  speed  in  revolutions  per  minute  and  the 
actual  lift,  the  efficiency  being  .759. 

Ans.  23.695  ft.  per  second  ;  ft  =  14°  44' ;  73.13  cu.  ft.  ;  80.13  : 
19.34  ft. 

22.  A  centrifugal  pump,  with  an  actual  lift  of  10  ft.,  delivers  37.85  cu. 
ft.  of  water  per  second  at  a  speed  of  68  revolutions  per  minute.     The 
number  of  blades  =  6  ;  the  blade-thickness  =  £•  in.;  the  wheel-depth  at 
outlet  =  9  ins.  ;  the  diameters  of  the  suction-  and  discharge-pipes  =  36 
ins.;  the  diameter  of  the  wheel  =  90  ins.  ;  0  =  19°  7'  26.67"  ;    Y  =  90°. 
Find  the  gain  of  head  in  passing  from  the  wheel  into  the  volute  and  the 
frictional  loss  (h^)  in  the  discharge-  and  suction-pipes  and  in  the  wheel- 
passages.     Also  find  the  efficiency. 

Ans.  2.193  ft-  ;  1.911  ft.;  .65. 

23.  In  the    centrifugal    pumps  for   two  torpedo-boat  destroyers  the 
diameter  of  eye  =  7  ins.  ;  the  diameter  of  wheel  =  20  ins.  ;  the  number 
of  blades  =  6  ;  the  thickness  of  blades  =  T\  in.;  the  width  of  the  wheel 
at  outlet  =  !-&  ins.;   the   actual  lift  =  63^  ins.;   cot  ft  =  5.167.     The 
pumps  are  driven   by  a  vertical    non-condensing  engine   with  a  4|-in. 
cylinder,  a  4-111.  stroke,  and  a  $-in.  piston-rod.     With  a  boiler-pressure 


57  EXAMPLES. 

of  220  Ibs.  per  square  inch  above  the  atmosphere  and  a  cut-off  at  f, 
the  delivery  was  found  to  be  1113  gallons  (U.  S.)  at  420  revolutions 
per  minute.  The  frictional  losses,  due  to  one  upper  bend,  two  7-111. 
bends,  one-  bad  check-valve,  one  gate-valve,  and  about  8  ft.  of  7-in. 
pipe,  were  respectively  estimated  at  .3/14,  .^hd,  /id,  .i/id,  and  .4185  ft., 
/id  being  the  head  corresponding  to  the  velocity  of  discharge  (=  velocity 
of  flow  in  volute).  Find  (a)  the  mechanical  efficiency;  and  also  find, 
on  the  ordinary  hypotheses  and  assuming  y  =  90°,  (b)  the  radial  velocity 
of  flow  ;  (c)  the  loss  in  shock  on  entering  the  volute ;  (d)  the  hydraulic 
efficiency. 
Ans.  (a)  6.02  per  cent ;  (b)  4.014  ft.  per  sec. ;  (c)  61.7  ft.-lbs. ;  (d)  .434. 

24.  Show  how  the  results  in  the  preceding  example  will  be  affected 
with  a  delivery  of  2000  U,  S.  gallons  at  an  assumed  speed  of  700  revo- 
lutions per  minute. 

Ans.  (a]  6.67  per  cent ;  (b)  7.214  ft.  per  sec. ;  (c)  120  ft.-lbs.  ;  (d)  .409. 

25.  Determine    the    hypothetically   best   speeds    in    revolutions    per 
minute  for  the  pumps  in  Examples  23  and  24,  and  calculate  the  corre- 
sponding maximum  hydraulic  efficiencies. 

Ans.  In  Ex.  23  best  speed  =  292.7  rev.  per  min. 
"     "     24     "         "      =  526       "      "      " 

26.  A  centrifugal  pump  delivers  20  cu.  ft.  of  water  per  second  at  a 
speed  of  225  revolutions  per  minute  ;  the  diameter  of  the  discharge-pipe 
is  20  ins.,  the  diameter  of  the  wheel  is  36  ins.;  the  width  of  the  wheel  at 
outlet  is  $$  ins.;  the  number  of  blades  =  6;  the  blade  thickness  =  f  in.; 
^  =  90°;  cosec  ft  —  1.484.     Find  the  hydraulic  efficiency,  and  also  find 
the  diameter  of  the  whirlpool-chamber  which  will  increase  this  efficiency 

by  .1234.  Ans.  .708 — ^-  ;  48  ins. 

27.  A  centrifugal  pump  making  229^  revolutions  per  minute  delivers 
23^  cu.  ft.  of  water  per  second.     The  diameter  of  the  discharge-pipe  = 
18  ins.,  of  the  wheel  =  42  ins.,  and  of  its  whirlpool-chamber  =48  ins. 
The  width  of  the  wheel  at  outlet  =  3.452  ins.,  and  of  the  whirlpool-cham- 
ber at  its  outer  circumference  =  2.5  ins.     The  tip  angle  ft  at  outlet  = 
cot~J  3.6.     Assuming  the  ordinary  whirlpool  theory  and  disregarding 
hydraulic  resistances,  determine  (a)  the  radial   velocity  of  flow   (vr")\ 
(b)  the  actual  velocity,  z/2,  with  which  the  water  leaves  the  wheel;  (c}  the 
loss  in  entering  the  whirlpool-chamber ;    (d}   the   hydraulic  efficiency. 
There  are  six  blades  each  f  in.  thick. 

Ans.  (a]  9.3569  ft.  per  sec.;  (b)  12.567  ft.  per  sec.; 
(c)  76.4142  ft.-lbs.;         (d)  .49. 


INDEX. 


Abbot,  252,  253,  257 

Abrupt  changes  of  section,  loss  of 

head  due  to,  164 
Accumulators,  339 
Accumulator,  Brown's  steam,  344 

differential,  342 
Air  in  a  pipe,  183 
Air,  retarding  effect  of,  224 
Applications   of   Bernouilli's   Theo- 
rem, 12 
Aqueducts,  circular,  242 

egg-shaped,  244 

flow-in,  240 

square,  243 
Arc  of  discharge  in  overshot  wheel, 

452 

Aspirator,  16 
Axial-flow  turbine,  490 

Balancing  of  hoists,  345 

Barker's  mill,  375 

Barlow's  curve,  73 

Barnes,  130 

Barometer,  water,  7 

Bazin,  230,  246,  248,  249,  250,  252, 

257,  258,  260,  266 
Bazin's  velocity  curve  and  formula, 

265, 266 

Bazin's  weirs,  99 
Bear,  punching,  339 
Beardmore,  247 
Beaufoy,  122 
Belgrand,  226 

Belgrand's  sewer  formula,  246 
Belidor,  386 
Bellmouth,  36 
Bends  in  pipe,  168 
Bends,  river,  269 
Bernouilli's  Theorem,  8 

applications  of,  12 
Bidone,  60,  284 


Binding-press,  338 
Boileau,  268 

Boileau's    velocity    curve    and    for- 
mula, 268 
Borda,  60 

Borda's  mouthpiece,  58 
Borda's  turbine,  382 
Bordered  vane,  368 
Bossut,  418 
Bourgogne   canal,   experiments  on, 

249.  257 

Bovey's  tables  of  coefficients  of  dis- 
charge, 39,  40 
Boyden's  hook  gauge,  298 
Boyden's  diffusor,  492 
Brakes,  hydraulic,  353 
Bramah's  press,  336 
Branched     pipe     connecting     three 

reservoirs,  191 
Branch    main  of  uniform  diameter, 

188 

Breadth  of  water-wheels,  438 
Breast-wheels,  440 
Breast-wheel,  efficiency  of,  441 

losses  of  effect  in,  442 

mechanical  effect  of,  442 

speed  of,  441 
Bresse,  291,  292,  296,  309 
Broad-crested  weir,  94 
Brotherhood  hydraulic  engine,  345. 
Brown's  steam-accumulator,  344 
Brumings,  247 
Bucket,  capacity  of,  458 

form  of,  435,  458 
Buckets,  number  of,  458 
Burdin's  wheel,  385 

Canal-lock,   time  of  emptying    and 

filling  a;  59 
Capacity    of    water-wheel    buckets, 

458 

577 


578 


INDEX. 


Capillary  phenomenon,  130 
Capillary  tubes,  flow  in,  130 
Castel's    table     of    mouthpiece    co- 
efficients, 69 
Centre  of  pressure,  xiv 
Centrifugal  force,  effect  of,  451 
Centrifugal  pump,  76 
analysis  of,  553 
efficiency  of,  555 
height  of  suction  in,  554 
losses  due  to  hydraulic  resistance 

in,  554 

values  of  a,  ft,  and  y  in,  556 
vortex  chamber  in,  565 
work  of,  394,  555 
Centrifugal  turbine,  393 
Chamber,  whirlpool,  76,  565 
Channel-flow  assumptions,  220 
Channel,    bottom    velocity    of    flow 

in  a,  266 

flow  between  bridge  piers  in  a,  296 
flow  in  an  open,  221 
flow  through    contracted    portion 

of  a,  293 
form  of,  228 
maximum  velocity    of  flow  in    a, 

236,  258, 265 

mean  velocity  of  flow  in  a,  268 
mid-depth  velocity  of  flow  in  a,  265 
of  great  width  as  compared  with 

the  depth,  288 
of  rectangular  section  and   small 

slope,  287 

steady  flow  in  a,  221 
surface  velocity  of  flow  in  a,  265 
value  of  tfand  ft  in  a,  249;  of  y  in 

a,  250;  of  n  in  a,  251 
variation   of  velocity  in  a  section 

of  a,  257 
Channels,  cycloidal,  239 

differential  equation  of  flow  in,  275 
examples  of,  228 
longitudinalprofile  of,  285 
of   constant    section,  steady   flow 

in,  271 

of  varying  section,  flow  in,  271 
rectangular,  229 
semi-circular,  238 
semi-elliptic,  239 
surface-slope  in,  227 
trapezoidal,  231 
with  change  of  section,  293 
with    constant    mean  velocity    of 

flow,  235 

Chezy's  formula,  163 
Chezy's  experiments  on  Courparlet 
channel,  247 


Circular  orifices,  81 
Cock  in  cylindrical  pipe,  169 
Cocks,  loss  of  head  due  to,  169 
Coefficients,  hydraulic,  29 
Coefficients  for  turbines,  519 
Coefficient  of  contraction,  34 

discharge,  38 

friction,  124 

resistance,  34 

velocity,  30 

viscosity,  269 
Coker,  130 

Combined-flow  turbines,  495 
Compressibility,  25 
Constants,  useful,  xvii 
Continuity,  27 
Contraction,  imperfect,  34 

incomplete,  35 

loss  of  head  due  to  abrupt,  165 
Coulomb,  122 
Courparlet  channel,  experiments  on, 

247 

Critical  velocity,  129 
Cunningham,  257 
Current-meters,  306 
Cylinders,  thickness  of,  337,  344 
Cylindrical  body  in   pipe,   pressure 

on,  406 
Cylindrical  mouthpiece,  63 

Danaides,  386 

Darcy,  126,  139,  249,  260.  303 

Darcy  gauge,  302 

D'Aubuisson,  126 

Defontaine's  velocity-curve  formula, 

262 

Density,  2 
Diagrams  of  pipe-flow  experiments, 

146 

Didion,  403 
Differential  accumulator,  342 

equation  of  steady  vaned  motion, 

275 

Diffusor,  Boyden's,  492 
Divergent  mouthpiece,  66 
Downward-flow  turbine,  490,  494 
Draft-tube,  theory  of,  529 
Drummond  on  Miner's  Inch,  44 
Dubiat,  247,  258,  403 
Dupuit,  293 

Efficiency    of     centrifugal     pumps, 

555 
Efficiency    of     turbines,    conditions 

governing,  510 
remarks  on,  519 
effect  of  centrifugal  force  on,  508 


INDEX. 


579 


Elasticity  of  volume,  6 
Elbows,  loss  of  head  due  to,  167 
Ellis,  177 

Energy,    losses    of    energy    in    hy- 
draulic machines,  351 

lost  in  shock,  55 

of  jet  of  water,  69 

of  water-fall,  7 

transmission  of,  156 

of  fluid,  kinetic,  n;  pressure,  n; 

weight,  II 
Engine,  hydraulic,  347 

speed  of  steady  motion  in,  351 
Enlargement  of  section,  loss  of  head 

due  to,  167 

Equations,  general,  53 
Equipotential  surface,  20 
Equivalent  uniform  main,  186 
Erosion  caused  by  watercourses,  227 

effect  of,  226,  227 

table  of,  269 
Examples,  109,  210,  328,355,408,539, 

572 

Exner.  308 

Expansion,  cubical,  6 
Experimental  tank,  29 
Eytelwein,  247,  248 

Farmer,  49,  81 

Flamant,  144 

Float    adjustment    in    experimental 

tank,  41 
Floats,  sub  surface,  300 

surface,  300 

twin,  301 
Flow  from  vessel  in  motion,  26 

in  a  frictionless  pipe,  27 

in  aqueducts,  240 

influence  of  pipe's  inclination  and 
position  upon  the,  138 

in  pipes,  133 

in  pipe  of  uniform  section,  133 

varying  diameter,  184 
Fluid,  definition  of,  xiii 

friction,  121 

motion,  i 

pressure,  xiii 

rotation,  17 

whirling  of,  19 
Foss,  143 

Fourneyron's  turbine,  491 
Fournie,  142 
Francis,  86,  89,  301 
Freeman,  178 
Free  surface,  20 
Friction,  coefficient  of.  124 

in  pipes,  surface,  125 


Friction,  laws  of  fluid,  123 
Frictionless  pipe,  flow  in,  27 
Froude,  131 

Froude's  table  of  frictional   resist- 
ances, 121 
Fteley,  89 
Funk,  247 

Ganguillet  &   Kutter's  formula,  250 

Gas,  definition  of,  xiii 

Ganges,  experiments  on,  257,  278 

Gauckler,  253 

Garonne,  experiments  on,  266 

Gauge,  Darcy,  303 

Hook,  298 
Gauging,  methods  of,  297 

of  pipe-flow,  207 

Gaugings  on  the  Ganges,  278;  Mis- 
sissippi, 253 
General  equations,  53 
Gerstner's  formula,  421 
Graphical  representation    of    losses 

of  head,  170 
Grashof,  431 
Grassi,  6 

Hagen,  139,  253 

Head,  27 

Hele  Shaw,  129 

Herschel,  208 

Hoists,  hydraulic  freight,  345 

Hook-gauge,  Boyden's,  298 

Humphreys,  252,  253,  257 

Hurdy-gurdy,  485 

Hydraulic  coefficients,  29 

engine,  344;  analysis  of,  347 

gradient,  13 

intensifier,  342 

jack,  338 
N   mean  depth,  222 

mean  radius,  135 

press,  335 

ram,  334 

Hydraulic  transmission,  156 
Hydraulics,  definition  of,  i 
Hydrodynamometer,  Perrodil's,  308 
Hydrometric  pendulum,  308 
Hydrostatics,      fundamental      prin- 
ciples of,  xiv 

Ice,  weight  of,  3 

Impact,  359 
apparatus,  369 
coefficient  of,  371 
on  a  flat  vane,  359 
on  a  curved  vane,  388 
on  a  hemispherical  vane,  367 
on  a  surface  of  revolution,  364 


INDEX. 


Impact  on  a  vane  with  borders,  368 
Impact  on  a  wheel,  378 
Imperfect  contraction,  34 
Inclination,   influence  of  pipe's,  138 
Injector,  15 
Intensifier,  341 
Inversion  of  the  jet,  48 
Inverted  siphon,  182 
Inward-flow  turbine,  490,  493 

Jack,  hydraulic,  338 
Jackson,  251 
Jet,  energy  of,  69 

inversion  of,  48 

measurer,  37 

momentum  of,  69 

propeller,  373 
Jet   reaction    wheel,    375;  efficiency 

of,  376;  useful  effect  of,  376 
Jet  turbine,  400 

Knibbs,  142 
Kutter,  142,  230,  253 

Laminar  motion,  2 

Lampe,  143 

Lesbros,  48 

Level  surface,  20 

Levy,  143 

Lift,  balanced  ram,  345 
hydraulic  ram,  346 

Limit  turbine,  494 

Lines  of  force,  20 

Liquid,  definition  of,  xiii 

Lock,  time  of  filling  a,  50 

Longitudinal   profile   of  open  chan- 
nel, 285 

Loss  of  energy  in  shock,  55 

Loss  of  head  due  to  abrupt  change 
of  section,  164  ;  bends.  168  ; 
cocks,  169  ;  contraction  of  sec- 
tion, 169  ;  elbows,  167  ;  enlarge-  j 
ment  of  section,  167  :  orifice  in 
diaphragm,  166 ;  sluices,  169  ; 
valves,  169 

Losses    of    head,    graphical  repre- 
sentation of,  170 

Losses  in  centrifugal  pumps,  559 
in  turbines,  531 

Magnus,  48 

Main,  equivalent  uniform,  186 
of  uniform  diameter,  branch,  188 
with  several  branches,  201 

Manning,  230,  252 

Mariotte,  403, 

Metacentre,  xv 


Meters,  207 

inferential,  209 

piston,  209 

rotary,  209 

Schonheyder's,  208 

Venturi,  207 
Meyer,  269 
Miner's  Inch,  44 
Mississippi,    experiments     on,    253, 

267 

Mixed-flow  turbines,  495 
Momentum  of  jet,  69 
Morin,  403,  431 
Motion,  fluid,  I 

in  plane  layers,  2 

in  stream-lines,  2 

laminar,  2 

permanent,  i 

steady,  I 
Motor  driven  by  water  flowing  along 

a  pipe,  179 
Mouthpiece,  Borda's,  58 

convergent,  66 

cylindrical,  63 

divergent,  66 

ring-nozzle,  61 

Navier's  hypothesis,  203,  264 
Notch,  83 

rectangular,  83 

triangular,    92 
Nozzles,  174 

Ellis'  experiments  on,  177 

Freeman's  experiments  on,  178 

Open  channels,  220 

Orifice  fed  by  two  reservoirs,  195 

flow  through  an,  23 

in  a  diaphragm,  loss  of  head  due 
to,  166 

in  a  thin  plate,  22 

in  vertical  plane  surfaces,  78 

in  vessel  in  motion,  26 

with  a  sharp  edge,  22 
Orifices,  circular,  81 

large,  78 

rectangular,  78 

semi-circular,  49 

triangular,  92 

Orleans  canal,  experiments  on,  247 
Outward-flow  turbine,  491 
Overshot-wheel,  450 

arc  of  discharge  in,  452 

bucket  angle  of,  456 

division  angle  in,  456 

effect  of  centrifugal  force  in,  451  ; 
impact  on,  469;  weight  on,  467 


INDEX. 


Overshot-wheel,  450 

number  of  buckets  in,  456,  458 
pitch-angle  in,  457 
speed  of,  450 
useful  effect  of,  467 
weight  of  water  on,  452 


Packing,  cup-leather,  336 

hemp,  336 

Parabolic  path  of  jet,  25 
Paraboloidal  surface,  20 
Paris  sewer  formula,  246 
Pastal's  press,  336 
Path  of  fluid  particle  in  turbine,  486 
Pelton  wheel,  486 
Pendulum,  hydrometric,  308 
Permanent  regime,  I 
Perrodil's  hydrodynamometer,  308 
Piezometer,  12 
Piobert,  403 

Pipe  connecting  three  reservoirs, 
branched,  191,  200  ;  two  reser- 
voirs, 162 

equivalent  uniform,  186 
flow  assumptions,  133 
flow  diagrams,  144 
flow  in  frictionless,  27 
Williams'  experiments  on  flow  in, 

206 

of  uniform  section,  flow  in,  133 
of  varying  section,  184 
thickness  of,  158,  159 
variation  of  velocity  in  transverse 

section  of,  202 
Pipe-flow,  effect  of  inclination  on, 

138 

Pitch-back  wheel,  472 
Pitot  tube,  302 
Plane  layers,  motion  in,  2 
Poiseuille,  128,  131 
Poncelet,  48,  418 

Poncelet  wheel,  424;  design  of,  433 
Position,  influence  of  pipe's,  137 
Practical  coefficients  in  centrifugal 

pumps,  569;  turbines,  519 
Piess,  Bramah's,  336 
Baling,  337 
hydraulic,  336 
Pressure,  centre  of,  xiv 

due  to  shock,  160 
Pressure-head,  n 

on  cylindrical  body  in  pipe,  406 
Pressure  on  thin  plate  in  pipe,  404 

of  fluids,  xiii 
Prony,  246,  248,  259 
Propeller,  jet,  373 


Pumps,    centrifugal,    547;    analysis 
of,  553;  vortex-chamber  in,  565 
Punching  bear,  339 

Radiating  current,  72 
Ram,  hydraulic,  335 
Rayleigh,  Lord,  48 
Reaction,  373 

Reaction  wheel,  efficiency  of,  376 
Rectangular  orifices,  78 
Regime,  permanent,  i 
Reservoir  sluices,  97 
Reservoirs,  branched  pipe  connect- 
ing three,  191,  200 

orifice  fed  by  two,  195 

pipe  connecting  two,  162 
Resistance  of  ships,  131 

of  motion  of  solids,  402 
Retarding  effect  of  air,  etc.,  in  chan- 
nel flow,  224 
Revy's  meter,  306 
Reynolds,  129,  130,  139,  141 
Rhine,  experiments  on,  247,  262,  266 
Ring-nozzle,  61 
River-bends,  269 
Riveter,  portable,  338 
Rotation  of  fluids,  17 
Riihlmann,  285,  286,  293 

Sagebien  wheels,  449 
Saone,  experiments  on,  257 
Schiele  turbine,  208 
Schonheyder's  meter,  257,  266 
Segner,  375 

Seine,  experiments  on,  257,  266 
Sharp-edge  orifices,  22 
Ships,  resistance  of,  131 
Shock,  energy  due  to,  55 

loss  of  energy  in,  55 

pressure  due  to,  160 
Simpson's  rule,  309 
Siphon,  181 

inverted,  182 
Slotte,  269 
Sluice  in  cylindrical  pipe,  169 

in  rectangular  pipe,  169 

loss  of  head  due  to  a,  169 
Sluices,  437 

reservoir,  97 

Smith,  Hamilton,  Jun.,  87 
Snow,  weight  of,  3 
Sonnet,  260 
Specific  gravity,  xiii 
Spiral  flow  of  water,  75 
Standing  wave,  281 
Steady  flow  in  channels  of  constant 
section,  221 


INDEX. 


Steady    motion,  i  ;  in    pipe  of    uni- 
form section,  133 
Steady    varied    motion,    differential 

equation  of,  202 
Stearns,  89 

Storage  of  energy,  340 
Stream  line,  2 
Strickland,  49 
St.  Venant,  248 
Suction-tube,  theory  of,  529 
Surface-floats,  300 
Surface-friction  in  pipes,  126 

slope  in  channels,  226 

tension,  49 

velocity,  258-265 


Tables  of  backwater  function,  290, 

291,  292 

bottom  velocities  269 
Castel's  results,  69 
coefficients  of  discharge,  39,  40 
coefficients   of  weir  discharge  by 

Fteley  &  Stearns,  89 
density  of  water,  4 
discharge  through  Miner's  Inch,  46 
discharge    through    nozzles,    177, 

178 

elasticity  of  volume  of  water,  6 
erosion  and  viscosity,  269 
expansion  of  volume  of  water,  6 
expansion  of  water,  4 
frictional  losses  in  hose,  178 
maximum  velocities,  269 
c  and  y  in  v  —  cmxti,  153 
showing  best  relative  dimensions 

for  trapezoidal  section,  233 
slopes  and  mean  velocities,  227 
slopes  of.  trapezoidal  section,  231 
values  of  c  and  b  in  Bazin's  form- 

ulae, 311  to  322 

values   of  •  —  -^—    for   centrifugal 


pumps,  556 
values  of  y  in  Bazin's  formula,  250 

TJ1  r 

values  of  —  —  for  turbines,  503 

Vg& 
values  of  m  and  n  in  Q=  m(h-\-  n), 

316 
values  of  n  in  Ganguillet  &   Kut- 

ter's  formula,  251 
viscosity  of    water  and    mercury, 

269 
values  of  c  and  b  in  channel  form- 

ulae, Ganguillet  &  Kutter,  323- 

326 


Tables,  values  of  c   and  b  in  Man- 
ning's formula,  327 
Tachometer,  308 
Tadini,  248 

Tank,  experimental,  29 
Tension,  surface,  49 
Theory  of  suction  or  draft  tube,  529 

of  turbines,  497 
Thjbault,  403 
Thickness  of   hydraulic    pipes   and 

cylinders,  337-344 
Thomson,  James,  77,  93,  269 
Thomson's  turbine,  565 
Throttle-valve,  loss  of  head  due  to, 

169 

Thrupp,  139 
Time    of     emptying     and    filling    a 

canal-lock,  50 
Torricelli's  theorem,  24 
Transmission  of  energy  by  hydrau- 
lic pressure,  136 
Trautwine,  89 
Tru  ngular  notch,  92 
Tub-wheel,  387 
Turbine,  axial-flow,  490,  494 

Borda's,  382 

Boyden's,  491 

centrifugal,  393 

combined,  495 

efficiency  of,  501,  510,  519 

Fontaine's,  494 

Fourneyron,  491 

impulse  or  Girard,   482,  507,  513, 

517 

inward-flow,  491,  493 
jet,  400 
Jonval,  494 
limit,  494 

losses  of  effect  in,  531 
mixed-flow,  490,  495 
outward-flow,  491 
parallel-flow,  494 
practical  values  of  velocities  in, 

519 

radial-flow,  490 
reaction,  482,  516 
Schiele,  495 
Scotch,  375 
Segner,  376 
Swain's,  495 
tangential,  393 
theory  of,  497 
Thomson,  491,  493 
useful  work  of,  501 
ventilated   483 
vortex,  49r,  493 
Whitelaw,  375 


INDEX. 


583 


Tutton,  146,  253,  289,  291,  292 
Tweddell's  differential  accumulator, 
342 

Undershot-wheel,  416 
Undershot  wheel,  actual  delivery  in 
ft.-lbs.  of,  423 

depth  of  crown  of,  431 

efficiency  of,  417,420;  Poncelet,  428 

form  of  course  of,  429 

in  a  straight  race,  418 

losses  of  effect  with,  421 

modifications  to  increase  efficiency 
of,  423 

number  of  buckets  in,  419 

Poncelet's,  424;  efficiency  of,  428 

useful  work  of,  417,  420 

with  flat  vanes,  417 
Uniform  main,  equivalent,  186 
Unwin,  403 
Useful  constants,  xvii 

Vallot,   143 

Values  of  c,  x,  and  y  in  v  =  cmxiv, 

153 

Valve,  loss  of  head  due  to  a,  169 
Vane,  best  form  of,  388 

cup,  367 
Velocity,  bottom,  260,  266 

critical,  129 

curve  in  a  channel,  257 

formulae,  Bazin's,  266 

formulae,  Boileau's,  268 

maximum,  260,  267 

mean,  258,  265 

mid-depth,  265 

of  whirl,  498 

rod,  301 

surface,  258,  265 

variation  of,  257 
Velocities     in     turbines,     practical 

values  of,  519 
Vena  contracta,  23 
Venant,  St.,  248 
Ventilated  buckets,  472 
Venturi,  water-meter,  16 
Vessels  in  motion,  orifice  in,  26 
Virtual  fall,  13 

slope,  13 
Viscosities,  table  of,  269 


Viscosity,  264 

Meyer's  formula  for,  269 

Slotte's  formula  for,  269 
Volute  of  centrifugal  pump,  558 
Vortex,  circular,  74 

compound,  76 

free,  74 

free-spiral,  75 

forced,  75 

motion,  74 

Water,  pressure  of,  6   ) 

weight  of,  2 
Water-barometer,  7 
Water-meter,  207 
Water-pressure  engine,  347 
Water-wheels,  classification  of  ver- 
tical, 416 

Wave  propagation,  velocity  of,  161 
Weight  of  fresh  water,  3 

of  ice,  3 

of  salt  water,  3 
Weir,  83 

Bazin's  flow-over,  99 

Beam,  104,   107 

broad-crested,  94,  106 

drowned,  88,  106 

inclined,  89 

rectangular,  with  end  contrac- 
tions, 86  ;  without  end  contrac- 
tions, 85 

sharp-crested,  99,  107 

submerged,  88 
Weisbach,  36,  60,  166 
Weser,  experiments  on,  247 
Wheel,  breast,  440 

hurdy-gurdy,  485 

in  straight  race,  418 

jet  reaction,  375 

overshot,  450 

Pelton,  486 

pitch-back,  472 

Poncelet,  424 

Sagebien,  449 

undershot,  416 
Whirling  fluids,  19 
Whirlpool-chamber,  76 
Whirl,  velocity  of,  519  ., 
Whitelaw,  375 
Williams,  206 
Woltmann,  247 


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